.. title:: COCO: The Bi-objective Black-Box Optimization Benchmarking (bbob-biobj) Test Suite $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ COCO: The Bi-objective Black-Box Optimization Benchmarking (``bbob-biobj``) Test Suite $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ .. the next two lines are necessary in LaTeX. They will be automatically replaced to put away the \chapter level as ??? and let the "current" level become \section. .. CHAPTERTITLE .. CHAPTERUNDERLINE .. | .. | .. .. sectnum:: :depth: 3 :numbered: .. .. contents:: Table of Contents :depth: 2 .. | .. | .. raw:: html See also: ArXiv e-prints, arXiv:1604.00359, 2016. .. raw:: latex % \tableofcontents TOC is automatic with sphinx and moved behind abstract by swap...py \begin{abstract} Several test function suites for numerical benchmarking of multiobjective optimization algorithms have been proposed in recent years. While having desirable properties like well-understood Pareto sets and Pareto fronts with shapes of various kinds, most of the currently used functions posess properties which are arguably under-represented in real-world problems. Those properties mainly stem from the easier construction of such problems---overrepresenting properties such as no dependencies between variables, Pareto sets exactly located at the bound constraints, or the differentiation between position and distance variables. Here, we propose an alternative way and define the ``bbob-biobj`` test suite with 55 bi-objective functions and its extended ``bbob-biobj-ext`` version with 92 bi-objective functions in continuous domain which are both derived from combining functions of the well-known single-objective noiseless ``bbob`` test suite. Besides giving the actual function definitions and presenting their (known) properties, this documentation also aims at giving the rationale behind our approach in terms of function groups, instances, and potential objective space normalization. .. raw:: latex \end{abstract} \newpage .. old
The bbob-biobj
test suite contains 55 bi-objective
functions in continuous domain which are derived from combining functions
of the well-known single-objective noiseless bbob
test suite. It will be used
as the main test suite of the upcoming BBOB-2016 workshop
at GECCO. Besides giving the actual function definitions and presenting
their (known) properties, this documentation also aims at giving the
rationale behind our approach in terms of function groups, instances, and
objective space normalization.
bbob-biobj
function 1 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_1$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f01-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f01-i01-d05-searchspace.*
:width: 49%
.. |f01-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f01-i01-d05-searchspace-projection.*
:width: 49%
.. |f01-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f01-i01-d05-logobjspace.*
:width: 49%
.. |f01-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f01-i01-d05-objspace.*
:width: 49%
.. _f2:
:math:`f_2`: Sphere/Ellipsoid separable
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the separable ellipsoid function (|f`2` in the bbob suite|_).
Both objectives are unimodal and separable. While the first objective is
truly convex-quadratic with a condition number of 1, the second
objective is only globally quadratic with smooth local
irregularities and highly ill-conditioned with a condition number of
about :math:`10^6`.
Contained in the *separable - separable* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_1`: Is symmetry exploited?
|f02-i01-d05-searchspace| |f02-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_2$ in dimension 5 for the first instance.\\[1em]
|f02-i01-d05-logobjspace| |f02-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 2 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_2$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f02-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f02-i01-d05-searchspace.*
:width: 49%
.. |f02-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f02-i01-d05-searchspace-projection.*
:width: 49%
.. |f02-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f02-i01-d05-logobjspace.*
:width: 49%
.. |f02-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f02-i01-d05-objspace.*
:width: 49%
.. _f3:
:math:`f_3`: Sphere/Attractive sector
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the attractive sector function (|f`6` in the bbob suite|_).
Both objective functions are unimodal, but only the first objective is
separable and truly convex quadratic. The attractive sector
function is highly asymmetric, where only one *hypercone* (with
angular base area) with a volume of roughly :math:`(1/2)^n`
yields low function values. The optimum of it is located at the tip
of this cone.
Contained in the *separable - moderate* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_1` and :math:`f_{20}`: What is the
effect of a highly asymmetric landscape in both or one
objective?
|f03-i01-d05-searchspace| |f03-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_3$ in dimension 5 for the first instance.\\[1em]
|f03-i01-d05-logobjspace| |f03-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 3 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_3$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f03-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f03-i01-d05-searchspace.*
:width: 49%
.. |f03-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f03-i01-d05-searchspace-projection.*
:width: 49%
.. |f03-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f03-i01-d05-logobjspace.*
:width: 49%
.. |f03-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f03-i01-d05-objspace.*
:width: 49%
.. _f4:
:math:`f_4`: Sphere/Rosenbrock original
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the original, i.e., unrotated Rosenbrock function (|f`8` in the
bbob suite|_).
The first objective is separable and truly convex, the second
objective is partially separable (tri-band structure). The first
objective is unimodal while the second objective has a local
optimum with an attraction volume of about 25\%.
Contained in the *separable - moderate* function class.
.. .. rubric:: Information gained from this function:
.. * Can the search follow a long path with :math:`n-1` changes in
the direction when it approaches one of the extremes of the
Pareto front/Pareto set?
|f04-i01-d05-searchspace| |f04-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_4$ in dimension 5 for the first instance.\\[1em]
|f04-i01-d05-logobjspace| |f04-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 4 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_4$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f04-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f04-i01-d05-searchspace.*
:width: 49%
.. |f04-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f04-i01-d05-searchspace-projection.*
:width: 49%
.. |f04-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f04-i01-d05-logobjspace.*
:width: 49%
.. |f04-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f04-i01-d05-objspace.*
:width: 49%
.. _f5:
:math:`f_5`: Sphere/Sharp ridge
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the sharp ridge function (|f`13` in the bbob suite|_).
Both objective functions are unimodal.
In addition to the simple, separable, and differentiable first
objective, a sharp, i.e., non-differentiable ridge has to be
followed for optimizing the (non-separable) second objective. The
gradient towards the ridge remains constant, when the ridge is
approached from a given point.
Approaching the ridge is initially effective, but becomes ineffective
close to the ridge when the rigde needs to be followed in direction
to its optimum. The necessary change in *search behavior* close to
the ridge is difficult to diagnose, because the gradient
towards the ridge does not flatten out.
Contained in the *separable - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * Can the search continuously change its search direction when
approaching one of the extremes of the Pareto front/Pareto set?
.. * What is the effect of having a non-smooth, non-differentiable
function to optimize?
|f05-i01-d05-searchspace| |f05-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_5$ in dimension 5 for the first instance.\\[1em]
|f05-i01-d05-logobjspace| |f05-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 5 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_5$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f05-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f05-i01-d05-searchspace.*
:width: 49%
.. |f05-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f05-i01-d05-searchspace-projection.*
:width: 49%
.. |f05-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f05-i01-d05-logobjspace.*
:width: 49%
.. |f05-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f05-i01-d05-objspace.*
:width: 49%
.. _f6:
:math:`f_6`: Sphere/Sum of different powers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the sum of different powers function (|f`14` in the bbob suite|_).
Both objective functions are unimodal. The first objective is
separable, the second non-separable.
When approaching the second objective's optimum, the difference
in sensitivity between different directions in search space
increases unboundedly.
.. In addition, the second objective function
possesses a small solution volume.
Contained in the *separable - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
|f06-i01-d05-searchspace| |f06-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_6$ in dimension 5 for the first instance.\\[1em]
|f06-i01-d05-logobjspace| |f06-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 6 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_6$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f06-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f06-i01-d05-searchspace.*
:width: 49%
.. |f06-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f06-i01-d05-searchspace-projection.*
:width: 49%
.. |f06-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f06-i01-d05-logobjspace.*
:width: 49%
.. |f06-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f06-i01-d05-objspace.*
:width: 49%
.. _f7:
:math:`f_7`: Sphere/Rastrigin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the Rastrigin function (|f`15` in the bbob suite|_).
In addition to the simple sphere function, the prototypical highly
multimodal Rastrigin function needs to be solved which has originally
a very regular and symmetric structure for the placement of the optima.
Here, however, transformations are performed to alleviate
the original symmetry and regularity in the second objective.
The properties of the second objective contain non-separabilty,
multimodality (roughly :math:`10^n` local optima), a conditioning of
about 10, and a large global amplitude compared to the local amplitudes.
Contained in the *separable - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * With respect to fully unimodal functions: what is the effect of
multimodality?
|f07-i01-d05-searchspace| |f07-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_7$ in dimension 5 for the first instance.\\[1em]
|f07-i01-d05-logobjspace| |f07-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 7 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_7$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f07-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f07-i01-d05-searchspace.*
:width: 49%
.. |f07-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f07-i01-d05-searchspace-projection.*
:width: 49%
.. |f07-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f07-i01-d05-logobjspace.*
:width: 49%
.. |f07-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f07-i01-d05-objspace.*
:width: 49%
.. _f8:
:math:`f_8`: Sphere/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the Schaffer F7 function with condition number 10 (|f`17` in
the bbob suite|_).
In addition to the simple sphere function, an asymmetric, non-separable,
and highly multimodal function needs to be solved to approach the Pareto
front/Pareto set where the frequency and amplitude of the modulation
in the second objective vary. The conditioning of the second objective
and thus the entire bi-objective function is low.
Contained in the *separable - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_7` and :math:`f_{50}`: What is the
effect of multimodality on a less regular function?
|f08-i01-d05-searchspace| |f08-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_8$ in dimension 5 for the first instance.\\[1em]
|f08-i01-d05-logobjspace| |f08-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 8 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_8$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f08-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f08-i01-d05-searchspace.*
:width: 49%
.. |f08-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f08-i01-d05-searchspace-projection.*
:width: 49%
.. |f08-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f08-i01-d05-logobjspace.*
:width: 49%
.. |f08-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f08-i01-d05-objspace.*
:width: 49%
.. _f9:
:math:`f_9`: Sphere/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the Schwefel function (|f`20` in the bbob suite|_).
While the first objective function is separable and unimodal,
the second objective function is partially separable and highly
multimodal---having the most prominent :math:`2^n` minima located
comparatively close to the corners of the unpenalized search area.
Contained in the *separable - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to e.g. :math:`f_8`: What is the effect of a weak
global structure?
|f09-i01-d05-searchspace| |f09-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_9$ in dimension 5 for the first instance.\\[1em]
|f09-i01-d05-logobjspace| |f09-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 9 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_9$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f09-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f09-i01-d05-searchspace.*
:width: 49%
.. |f09-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f09-i01-d05-searchspace-projection.*
:width: 49%
.. |f09-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f09-i01-d05-logobjspace.*
:width: 49%
.. |f09-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f09-i01-d05-objspace.*
:width: 49%
.. _f10:
:math:`f_{10}`: Sphere/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sphere function (|f`1` in the bbob suite|_)
and the Gallagher function with 101 peaks (|f`21` in the bbob
suite|_).
While the first objective function is separable and unimodal,
the second objective function is non-separable and consists
of 101 optima with position and height being unrelated and
randomly chosen (different for each instantiation of the function).
The conditioning around the global optimum of the second
objective function is about 30.
Contained in the *separable - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Is the search effective without any global structure?
|f10-i01-d05-searchspace| |f10-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{10}$ in dimension 5 for the first instance.\\[1em]
|f10-i01-d05-logobjspace| |f10-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 10 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{10}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f10-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f10-i01-d05-searchspace.*
:width: 49%
.. |f10-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f10-i01-d05-searchspace-projection.*
:width: 49%
.. |f10-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f10-i01-d05-logobjspace.*
:width: 49%
.. |f10-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f10-i01-d05-objspace.*
:width: 49%
.. _f11:
:math:`f_{11}`: Ellipsoid separable/Ellipsoid separable
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two separable ellipsoid functions (|f`2` in the
bbob suite|_).
Both objectives are unimodal, separable, only globally
quadratic with smooth local irregularities, and highly
ill-conditioned with a condition number of
about :math:`10^6`.
Contained in the *separable - separable* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_1`: Is symmetry (rather: separability) exploited?
|f11-i01-d05-searchspace| |f11-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{11}$ in dimension 5 for the first instance.\\[1em]
|f11-i01-d05-logobjspace| |f11-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 11 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{11}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f11-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f11-i01-d05-searchspace.*
:width: 49%
.. |f11-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f11-i01-d05-searchspace-projection.*
:width: 49%
.. |f11-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f11-i01-d05-logobjspace.*
:width: 49%
.. |f11-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f11-i01-d05-objspace.*
:width: 49%
.. _f12:
:math:`f_{12}`: Ellipsoid separable/Attractive sector
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the bbob suite|_)
and the attractive sector function (|f`6` in the bbob suite|_).
Both objective functions are unimodal but only the first
one is separable. The first objective function, in addition,
is globally quadratic with smooth local irregularities, and
highly ill-conditioned with a condition number of about
:math:`10^6`. The second objective function is highly
asymmetric, where only one *hypercone* (with
angular base area) with a volume of roughly :math:`(1/2)^n`
yields low function values. The optimum of it is located at
the tip of this cone.
Contained in the *separable - moderate* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to, for example, :math:`f_1`: Is symmetry exploited?
|f12-i01-d05-searchspace| |f12-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{12}$ in dimension 5 for the first instance.\\[1em]
|f12-i01-d05-logobjspace| |f12-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 12 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{12}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f12-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f12-i01-d05-searchspace.*
:width: 49%
.. |f12-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f12-i01-d05-searchspace-projection.*
:width: 49%
.. |f12-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f12-i01-d05-logobjspace.*
:width: 49%
.. |f12-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f12-i01-d05-objspace.*
:width: 49%
.. _f13:
:math:`f_{13}`: Ellipsoid separable/Rosenbrock original
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the
bbob suite|_) and the original, i.e., unrotated Rosenbrock function
(|f`8` in the bbob suite|_).
Only the first objective is separable and unimodal. The second
objective is partially separable (tri-band structure) and has a local
optimum with an attraction volume of about 25\%.
In addition, the first objective function shows smooth local
irregularities from a globally convex quadratic function and is
highly ill-conditioned with a condition number of about
:math:`10^6`.
Contained in the *separable - moderate* function class.
.. .. rubric:: Information gained from this function:
.. * Can the search handle highly conditioned functions and follow a long
path with :math:`n-1` changes in the direction when it approaches the
Pareto front/Pareto set?
|f13-i01-d05-searchspace| |f13-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{13}$ in dimension 5 for the first instance.\\[1em]
|f13-i01-d05-logobjspace| |f13-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 13 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{13}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f13-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f13-i01-d05-searchspace.*
:width: 49%
.. |f13-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f13-i01-d05-searchspace-projection.*
:width: 49%
.. |f13-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f13-i01-d05-logobjspace.*
:width: 49%
.. |f13-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f13-i01-d05-objspace.*
:width: 49%
.. _f14:
:math:`f_{14}`: Ellipsoid separable/Sharp ridge
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the
bbob suite|_) and the sharp ridge function (|f`13` in the bbob suite|_).
Both objective functions are unimodal but only the first one is
separable.
The first objective is globally quadratic but with smooth local
irregularities and highly ill-conditioned with a condition number of
about :math:`10^6`. For optimizing the second objective, a sharp,
i.e., non-differentiable ridge has to be followed.
Contained in the *separable - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * Can the search continuously change its search direction when
approaching one of the extremes of the Pareto front/Pareto set?
.. * What is the effect of having to solve both a highly-conditioned
and a non-smooth, non-differentiabale function to approximate
the Pareto front/Pareto set?
|f14-i01-d05-searchspace| |f14-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{14}$ in dimension 5 for the first instance.\\[1em]
|f14-i01-d05-logobjspace| |f14-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 14 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{14}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f14-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f14-i01-d05-searchspace.*
:width: 49%
.. |f14-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f14-i01-d05-searchspace-projection.*
:width: 49%
.. |f14-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f14-i01-d05-logobjspace.*
:width: 49%
.. |f14-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f14-i01-d05-objspace.*
:width: 49%
.. _f15:
:math:`f_{15}`: Ellipsoid separable/Sum of different powers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the
bbob suite|_) and the sum of different powers function
(|f`14` in the bbob suite|_).
Both objective functions are unimodal but only the first one is
separable.
The first objective is globally quadratic but with smooth local
irregularities and highly ill-conditioned with a condition number of
about :math:`10^6`. When approaching the second objective's optimum,
the sensitivies of the variables in the rotated search space become
more and more different.
Contained in the *separable - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * Can the Pareto front/Pareto set be approached when both a
highly conditioned function and a function, the conditioning
of which increases when approaching the optimum, must be solved?
|f15-i01-d05-searchspace| |f15-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{15}$ in dimension 5 for the first instance.\\[1em]
|f15-i01-d05-logobjspace| |f15-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 15 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{15}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f15-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f15-i01-d05-searchspace.*
:width: 49%
.. |f15-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f15-i01-d05-searchspace-projection.*
:width: 49%
.. |f15-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f15-i01-d05-logobjspace.*
:width: 49%
.. |f15-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f15-i01-d05-objspace.*
:width: 49%
.. _f16:
:math:`f_{16}`: Ellipsoid separable/Rastrigin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the
bbob suite|_) and the Rastrigin function (|f`15` in the bbob suite|_).
The objective functions show rather opposite properties.
The first one is separable, the second not. The first one
is unimodal, the second highly multimodal (roughly :math:`10^n` local
optima). The first one is highly ill-conditioning (condition number of
:math:`10^6`), the second one has a conditioning of about 10. Local
non-linear transformations are performed in both objective functions
to alleviate the original symmetry and regularity of the two
baseline functions.
Contained in the *separable - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * With respect to fully unimodal functions: what is the effect of
multimodality?
.. * With respect to low-conditioned problems: what is the effect of
high conditioning?
|f16-i01-d05-searchspace| |f16-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{16}$ in dimension 5 for the first instance.\\[1em]
|f16-i01-d05-logobjspace| |f16-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 16 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{16}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f16-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f16-i01-d05-searchspace.*
:width: 49%
.. |f16-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f16-i01-d05-searchspace-projection.*
:width: 49%
.. |f16-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f16-i01-d05-logobjspace.*
:width: 49%
.. |f16-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f16-i01-d05-objspace.*
:width: 49%
.. _f17:
:math:`f_{17}`: Ellipsoid separable/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the
bbob suite|_) and the Schaffer F7 function with condition number 10
(|f`17` in the bbob suite|_).
Also here, both single objectives possess opposing properties.
The first objective is unimodal, besides small local non-linearities symmetric,
separable and highly ill-conditioned while the second objective is highly
multi-modal, asymmetric, and non-separable, with only a low conditioning.
Contained in the *separable - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the opposing difficulties posed by the
single objectives when parts of the Pareto front (at the extremes, in the
middle, ...) are explored?
|f17-i01-d05-searchspace| |f17-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{17}$ in dimension 5 for the first instance.\\[1em]
|f17-i01-d05-logobjspace| |f17-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 17 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{17}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f17-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f17-i01-d05-searchspace.*
:width: 49%
.. |f17-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f17-i01-d05-searchspace-projection.*
:width: 49%
.. |f17-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f17-i01-d05-logobjspace.*
:width: 49%
.. |f17-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f17-i01-d05-objspace.*
:width: 49%
.. _f18:
:math:`f_{18}`: Ellipsoid separable/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the
bbob suite|_) and the Schwefel function (|f`20` in the bbob suite|_).
The first objective is unimodal, separable and highly ill-conditioned.
The second objective is partially separable and highly multimodal---having
the most prominent :math:`2^n` minima located comparatively close to the
corners of the unpenalized search area.
Contained in the *separable - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. .. todo::
Give some details.
|f18-i01-d05-searchspace| |f18-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{18}$ in dimension 5 for the first instance.\\[1em]
|f18-i01-d05-logobjspace| |f18-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 18 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{18}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f18-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f18-i01-d05-searchspace.*
:width: 49%
.. |f18-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f18-i01-d05-searchspace-projection.*
:width: 49%
.. |f18-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f18-i01-d05-logobjspace.*
:width: 49%
.. |f18-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f18-i01-d05-objspace.*
:width: 49%
.. _f19:
:math:`f_{19}`: Ellipsoid separable/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the separable ellipsoid function (|f`2` in the
bbob suite|_) and the Gallagher function with 101 peaks (|f`21` in the bbob suite|_).
While the first objective function is separable, unimodal, and
highly ill-conditioned (condition number of about :math:`10^6`),
the second objective function is non-separable and consists
of 101 optima with position and height being unrelated and
randomly chosen (different for each instantiation of the function).
The conditioning around the global optimum of the second
objective function is about 30.
Contained in the *separable - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Is the search effective without any global structure?
.. * What is the effect of the different condition numbers
of the two objectives, in particular when combined
to reach the middle of the Pareto front?
|f19-i01-d05-searchspace| |f19-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{19}$ in dimension 5 for the first instance.\\[1em]
|f19-i01-d05-logobjspace| |f19-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 19 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{19}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f19-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f19-i01-d05-searchspace.*
:width: 49%
.. |f19-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f19-i01-d05-searchspace-projection.*
:width: 49%
.. |f19-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f19-i01-d05-logobjspace.*
:width: 49%
.. |f19-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f19-i01-d05-objspace.*
:width: 49%
.. _f20:
:math:`f_{20}`: Attractive sector/Attractive sector
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two attractive sector functions (|f`6`
in the bbob suite|_).
Both functions are unimodal and highly asymmetric, where only one
*hypercone* (with angular base area) per objective with a volume of
roughly :math:`(1/2)^n` yields low function values. The objective
functions' optima are located at the tips of those two cones.
Contained in the *moderate - moderate* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_1` and :math:`f_{20}`: What is the
effect of a highly asymmetric landscape in both or one
objective?
|f20-i01-d05-searchspace| |f20-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{20}$ in dimension 5 for the first instance.\\[1em]
|f20-i01-d05-logobjspace| |f20-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 20 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{20}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f20-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f20-i01-d05-searchspace.*
:width: 49%
.. |f20-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f20-i01-d05-searchspace-projection.*
:width: 49%
.. |f20-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f20-i01-d05-logobjspace.*
:width: 49%
.. |f20-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f20-i01-d05-objspace.*
:width: 49%
.. _f21:
:math:`f_{21}`: Attractive sector/Rosenbrock original
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the attractive sector function (|f`6`
in the bbob suite|_) and the Rosenbrock function (|f`8` in the bbob suite|_).
The first function is unimodal but highly asymmetric, where only one
*hypercone* (with angular base area) with a volume of
roughly :math:`(1/2)^n` yields low function values (with the
optimum at the tip of the cone). The second
objective is partially separable (tri-band structure) and has a local
optimum with an attraction volume of about 25\%.
Contained in the *moderate - moderate* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of relatively large search space areas
leading to suboptimal values of the two objective
functions?
|f21-i01-d05-searchspace| |f21-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{21}$ in dimension 5 for the first instance.\\[1em]
|f21-i01-d05-logobjspace| |f21-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 21 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{21}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f21-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f21-i01-d05-searchspace.*
:width: 49%
.. |f21-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f21-i01-d05-searchspace-projection.*
:width: 49%
.. |f21-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f21-i01-d05-logobjspace.*
:width: 49%
.. |f21-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f21-i01-d05-objspace.*
:width: 49%
.. _f22:
:math:`f_{22}`: Attractive sector/Sharp ridge
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the attractive sector function (|f`6`
in the bbob suite|_) and the sharp ridge function (|f`13` in the bbob suite|_).
Both objective functions are unimodal and non-separable. The
first objective is highly asymmetric in the sense that only one
*hypercone* (with angular base area) with a volume of
roughly :math:`(1/2)^n` yields low function values (with the
optimum at the tip of the cone). For optimizing the second
objective, a sharp, i.e., non-differentiable ridge has to be followed.
Contained in the *moderate - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of assymmetries and non-differentiabilities
when approaching the Pareto front/Pareto set?
|f22-i01-d05-searchspace| |f22-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{22}$ in dimension 5 for the first instance.\\[1em]
|f22-i01-d05-logobjspace| |f22-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 22 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{22}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f22-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f22-i01-d05-searchspace.*
:width: 49%
.. |f22-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f22-i01-d05-searchspace-projection.*
:width: 49%
.. |f22-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f22-i01-d05-logobjspace.*
:width: 49%
.. |f22-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f22-i01-d05-objspace.*
:width: 49%
.. _f23:
:math:`f_{23}`: Attractive sector/Sum of different powers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the attractive sector function (|f`6`
in the bbob suite|_) and the sum of different powers function
(|f`14` in the bbob suite|_).
Both objective functions are unimodal and non-separable. The
first objective is highly asymmetric in the sense that only one
*hypercone* (with angular base area) with a volume of
roughly :math:`(1/2)^n` yields low function values (with the
optimum at the tip of the cone). When approaching the second
objective's optimum, the sensitivies of the variables in the
rotated search space become more and more different.
Contained in the *moderate - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of assymmetries and an increasing
conditioning in one objective function (sum of different
powers function) when approaching Pareto-optimal points?
|f23-i01-d05-searchspace| |f23-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{23}$ in dimension 5 for the first instance.\\[1em]
|f23-i01-d05-logobjspace| |f23-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 23 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{23}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f23-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f23-i01-d05-searchspace.*
:width: 49%
.. |f23-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f23-i01-d05-searchspace-projection.*
:width: 49%
.. |f23-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f23-i01-d05-logobjspace.*
:width: 49%
.. |f23-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f23-i01-d05-objspace.*
:width: 49%
.. _f24:
:math:`f_{24}`: Attractive sector/Rastrigin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the attractive sector function (|f`6`
in the bbob suite|_) and the Rastrigin function
(|f`15` in the bbob suite|_).
Both objectives are non-separable, and the second one
is highly multi-modal (roughly :math:`10^n` local
optima) while the first one is unimodal. Further
properties are that the first objective is highly
assymetric and the second has a conditioning of about 10.
Contained in the *moderate - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * With respect to fully unimodal and rather symmetric functions:
what is the effect of multimodality and assymmetry?
|f24-i01-d05-searchspace| |f24-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{24}$ in dimension 5 for the first instance.\\[1em]
|f24-i01-d05-logobjspace| |f24-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 24 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{24}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f24-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f24-i01-d05-searchspace.*
:width: 49%
.. |f24-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f24-i01-d05-searchspace-projection.*
:width: 49%
.. |f24-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f24-i01-d05-logobjspace.*
:width: 49%
.. |f24-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f24-i01-d05-objspace.*
:width: 49%
.. _f25:
:math:`f_{25}`: Attractive sector/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the attractive sector function (|f`6`
in the bbob suite|_) and the Schaffer F7 function with condition number 10
(|f`17` in the bbob suite|_).
Both objectives are non-separable and asymmetric.
While the first objective is unimodal, the second one is
a highly multi-modal function with a low conditioning where
frequency and amplitude of the modulation vary.
Contained in the *moderate - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of having to solve the relatively` simple, but
asymmetric first objective together with the highly multi-modal
second objective with less regularities when the Pareto front/Pareto
Pareto set is approached?
|f25-i01-d05-searchspace| |f25-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{25}$ in dimension 5 for the first instance.\\[1em]
|f25-i01-d05-logobjspace| |f25-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 25 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{25}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f25-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f25-i01-d05-searchspace.*
:width: 49%
.. |f25-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f25-i01-d05-searchspace-projection.*
:width: 49%
.. |f25-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f25-i01-d05-logobjspace.*
:width: 49%
.. |f25-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f25-i01-d05-objspace.*
:width: 49%
.. _f26:
:math:`f_{26}`: Attractive sector/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the attractive sector function (|f`6`
in the bbob suite|_) and the Schwefel function (|f`20` in the bbob suite|_).
The first objective is non-separable, unimodal, and asymmetric.
The second objective is partially separable and highly multimodal---having
the most prominent :math:`2^n` minima located comparatively close to the
corners of the unpenalized search area.
Contained in the *moderate - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of asymmetries and a weak global structure when
different parts of the Pareto front/Pareto set are approached?
|f26-i01-d05-searchspace| |f26-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{26}$ in dimension 5 for the first instance.\\[1em]
|f26-i01-d05-logobjspace| |f26-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 26 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{26}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f26-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f26-i01-d05-searchspace.*
:width: 49%
.. |f26-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f26-i01-d05-searchspace-projection.*
:width: 49%
.. |f26-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f26-i01-d05-logobjspace.*
:width: 49%
.. |f26-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f26-i01-d05-objspace.*
:width: 49%
.. _f27:
:math:`f_{27}`: Attractive sector/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the attractive sector function (|f`6`
in the bbob suite|_) and the Gallagher function with 101 peaks (|f`21` in the bbob suite|_).
Both objective functions are non-separable but only the first
is unimodal. The first objective function is furthermore asymmetric.
The second objective function has 101 optima with position and height
being unrelated and randomly chosen (different for each instantiation
of the function). The conditioning around the global optimum of the second
objective function is about 30.
Contained in the *moderate - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Is the search effective without any global structure?
.. * What is the effect of the different condition numbers
of the two objectives, in particular when combined
to reach the middle of the Pareto front?
|f27-i01-d05-searchspace| |f27-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{27}$ in dimension 5 for the first instance.\\[1em]
|f27-i01-d05-logobjspace| |f27-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 27 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{27}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f27-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f27-i01-d05-searchspace.*
:width: 49%
.. |f27-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f27-i01-d05-searchspace-projection.*
:width: 49%
.. |f27-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f27-i01-d05-logobjspace.*
:width: 49%
.. |f27-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f27-i01-d05-objspace.*
:width: 49%
.. _f28:
:math:`f_{28}`: Rosenbrock original/Rosenbrock original
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two Rosenbrock functions (|f`8` in the bbob suite|_).
Both objectives are partially separable (tri-band structure) and have
a local optimum with an attraction volume of about 25\%.
Contained in the *moderate - moderate* function class.
.. .. rubric:: Information gained from this function:
.. * Can the search follow different long paths with $n-1$ changes in the
direction when approaching the extremes of the Pareto front/Pareto set?
.. * What is the effect when a combination of the two paths have to
be solved when a point in the middle of the Pareto front/Pareto set
is sought?
|f28-i01-d05-searchspace| |f28-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{28}$ in dimension 5 for the first instance.\\[1em]
|f28-i01-d05-logobjspace| |f28-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 28 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{28}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f28-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f28-i01-d05-searchspace.*
:width: 49%
.. |f28-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f28-i01-d05-searchspace-projection.*
:width: 49%
.. |f28-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f28-i01-d05-logobjspace.*
:width: 49%
.. |f28-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f28-i01-d05-objspace.*
:width: 49%
.. _f29:
:math:`f_{29}`: Rosenbrock original/Sharp ridge
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rosenbrock function (|f`8` in the bbob suite|_) and the
sharp ridge function (|f`13` in the bbob suite|_).
The first objective function is partially separable (tri-band structure)
and has a local optimum with an attraction volume of about 25\%.
The second objective is unimodal and non-separable and, for
optimizing it, a sharp, i.e., non-differentiable ridge has to be followed.
Contained in the *moderate - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the opposing difficulties posed by the
single objectives when parts of the Pareto front (at the extremes, in the
middle, ...) are explored?
|f29-i01-d05-searchspace| |f29-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{29}$ in dimension 5 for the first instance.\\[1em]
|f29-i01-d05-logobjspace| |f29-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 29 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{29}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f29-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f29-i01-d05-searchspace.*
:width: 49%
.. |f29-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f29-i01-d05-searchspace-projection.*
:width: 49%
.. |f29-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f29-i01-d05-logobjspace.*
:width: 49%
.. |f29-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f29-i01-d05-objspace.*
:width: 49%
.. _f30:
:math:`f_{30}`: Rosenbrock original/Sum of different powers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rosenbrock function (|f`8` in the bbob suite|_) and the sum of different powers function
(|f`14` in the bbob suite|_).
The first objective function is partially separable (tri-band structure)
and has a local optimum with an attraction volume of about 25\%.
The second objective function is unimodal and non-separable. When
approaching the second objective's optimum, the sensitivies of the
variables in the rotated search space become more and more different.
Contained in the *moderate - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of having to follow a long path with $n-1$ changes
in the direction when optimizing one objective function and an increasing
conditioning when solving the other, in particular when trying to
approximate the Pareto front/Pareto set not close to their extremes?
|f30-i01-d05-searchspace| |f30-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{30}$ in dimension 5 for the first instance.\\[1em]
|f30-i01-d05-logobjspace| |f30-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 30 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{30}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f30-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f30-i01-d05-searchspace.*
:width: 49%
.. |f30-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f30-i01-d05-searchspace-projection.*
:width: 49%
.. |f30-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f30-i01-d05-logobjspace.*
:width: 49%
.. |f30-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f30-i01-d05-objspace.*
:width: 49%
.. _f31:
:math:`f_{31}`: Rosenbrock original/Rastrigin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rosenbrock function (|f`8` in the bbob suite|_) and the Rastrigin function
(|f`15` in the bbob suite|_).
The first objective function is partially separable (tri-band structure)
and has a local optimum with an attraction volume of about 25\%.
The second objective function is non-separable and
highly multi-modal (roughly :math:`10^n` local
optima).
Contained in the *moderate - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * With respect to fully unimodal functions:
what is the effect of multimodality?
|f31-i01-d05-searchspace| |f31-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{31}$ in dimension 5 for the first instance.\\[1em]
|f31-i01-d05-logobjspace| |f31-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 31 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{31}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f31-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f31-i01-d05-searchspace.*
:width: 49%
.. |f31-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f31-i01-d05-searchspace-projection.*
:width: 49%
.. |f31-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f31-i01-d05-logobjspace.*
:width: 49%
.. |f31-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f31-i01-d05-objspace.*
:width: 49%
.. _f32:
:math:`f_{32}`: Rosenbrock original/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rosenbrock function (|f`8` in the bbob suite|_) and the
Schaffer F7 function with condition number 10
(|f`17` in the bbob suite|_).
The first objective function is partially separable (tri-band structure)
and has a local optimum with an attraction volume of about 25\%.
The second objective function is non-separable, asymmetric, and
highly multi-modal with a low conditioning where
frequency and amplitude of the modulation vary.
Contained in the *moderate - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the different difficulties (in particular
the high multi-modality of the second objective) when approaching
the Pareto front/Pareto set, especially in the middle?
|f32-i01-d05-searchspace| |f32-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{32}$ in dimension 5 for the first instance.\\[1em]
|f32-i01-d05-logobjspace| |f32-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 32 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{32}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f32-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f32-i01-d05-searchspace.*
:width: 49%
.. |f32-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f32-i01-d05-searchspace-projection.*
:width: 49%
.. |f32-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f32-i01-d05-logobjspace.*
:width: 49%
.. |f32-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f32-i01-d05-objspace.*
:width: 49%
.. _f33:
:math:`f_{33}`: Rosenbrock original/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rosenbrock function (|f`8` in the bbob suite|_) and the
Schwefel function (|f`20` in the bbob suite|_).
Both objective functions are partially separable.
While the first objective function has a local optimum with an attraction
volume of about 25\%, the second objective function is highly
multimodal---having the most prominent :math:`2^n` minima located
comparatively close to the corners of its unpenalized search area.
Contained in the *moderate - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the different difficulties (in particular
the high multi-modality and weak global structure of the second
objective) when approaching the Pareto front/Pareto set,
especially in the middle?
.. * Can the partial separability of the two objectives be detected
and exploited?
|f33-i01-d05-searchspace| |f33-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{33}$ in dimension 5 for the first instance.\\[1em]
|f33-i01-d05-logobjspace| |f33-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 33 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{33}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f33-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f33-i01-d05-searchspace.*
:width: 49%
.. |f33-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f33-i01-d05-searchspace-projection.*
:width: 49%
.. |f33-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f33-i01-d05-logobjspace.*
:width: 49%
.. |f33-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f33-i01-d05-objspace.*
:width: 49%
.. _f34:
:math:`f_{34}`: Rosenbrock original/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rosenbrock function (|f`8` in the bbob suite|_) and
the Gallagher function with 101 peaks (|f`21` in the bbob suite|_).
The first objective function is partially separable, the second one
non-separable. While the first objective function has a local optimum
with an attraction volume of about 25\%, the second objective function
has 101 optima with position and height being unrelated and randomly
chosen (different for each instantiation of the function). The
conditioning around the global optimum of the second objective function
is about 30.
Contained in the *moderate - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Is the search effective without any global structure?
.. * How much does the multi-modality play a role when compared to
fully uni-modal functions?
|f34-i01-d05-searchspace| |f34-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{34}$ in dimension 5 for the first instance.\\[1em]
|f34-i01-d05-logobjspace| |f34-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 34 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{34}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f34-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f34-i01-d05-searchspace.*
:width: 49%
.. |f34-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f34-i01-d05-searchspace-projection.*
:width: 49%
.. |f34-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f34-i01-d05-logobjspace.*
:width: 49%
.. |f34-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f34-i01-d05-objspace.*
:width: 49%
.. _f35:
:math:`f_{35}`: Sharp ridge/Sharp ridge
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two sharp ridge functions (|f`13` in the bbob suite|_).
Both objective functions are unimodal and non-separable and, for
optimizing them, two sharp, i.e., non-differentiable ridges have to be
followed.
Contained in the *ill-conditioned - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of having to follow non-smooth, non-differentiabale
ridges?
|f35-i01-d05-searchspace| |f35-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{35}$ in dimension 5 for the first instance.\\[1em]
|f35-i01-d05-logobjspace| |f35-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 35 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{35}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f35-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f35-i01-d05-searchspace.*
:width: 49%
.. |f35-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f35-i01-d05-searchspace-projection.*
:width: 49%
.. |f35-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f35-i01-d05-logobjspace.*
:width: 49%
.. |f35-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f35-i01-d05-objspace.*
:width: 49%
.. _f36:
:math:`f_{36}`: Sharp ridge/Sum of different powers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sharp ridge function (|f`13` in the bbob suite|_) and the
sum of different powers function
(|f`14` in the bbob suite|_).
Both functions are uni-modal and non-separable.
For optimizing the first objective, a sharp, i.e., non-differentiable
ridge has to be followed.
When approaching the second objective's optimum, the sensitivies of the
variables in the rotated search space become more and more different.
Contained in the *ill-conditioned - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of having to follow a ridge when optimizing one
objective function and an increasing conditioning when solving the other,
in particular when trying to approximate the Pareto front/Pareto set not
close to their extremes?
|f36-i01-d05-searchspace| |f36-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{36}$ in dimension 5 for the first instance.\\[1em]
|f36-i01-d05-logobjspace| |f36-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 36 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{36}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f36-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f36-i01-d05-searchspace.*
:width: 49%
.. |f36-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f36-i01-d05-searchspace-projection.*
:width: 49%
.. |f36-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f36-i01-d05-logobjspace.*
:width: 49%
.. |f36-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f36-i01-d05-objspace.*
:width: 49%
.. _f37:
:math:`f_{37}`: Sharp ridge/Rastrigin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sharp ridge function (|f`13` in the bbob suite|_) and the Rastrigin function
(|f`15` in the bbob suite|_).
Both functions are non-separable. While the first one
is unimodal and non-differentiable at its ridge, the second objective
function is highly multi-modal (roughly :math:`10^n` local optima).
Contained in the *ill-conditioned - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of having to follow a ridge when optimizing one
objective function and the high multi-modality of the other,
in particular when trying to approximate the Pareto front/Pareto set not
close to their extremes?
|f37-i01-d05-searchspace| |f37-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{37}$ in dimension 5 for the first instance.\\[1em]
|f37-i01-d05-logobjspace| |f37-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 37 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{37}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f37-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f37-i01-d05-searchspace.*
:width: 49%
.. |f37-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f37-i01-d05-searchspace-projection.*
:width: 49%
.. |f37-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f37-i01-d05-logobjspace.*
:width: 49%
.. |f37-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f37-i01-d05-objspace.*
:width: 49%
.. _f38:
:math:`f_{38}`: Sharp ridge/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sharp ridge function (|f`13` in the bbob suite|_) and the
Schaffer F7 function with condition number 10
(|f`17` in the bbob suite|_).
Both functions are non-separable. While the first one
is unimodal and non-differentiable at its ridge, the second objective
function is asymmetric and highly multi-modal with a low conditioning where
frequency and amplitude of the modulation vary.
Contained in the *ill-conditioned - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the different difficulties when approaching
the Pareto front/Pareto set, especially in the middle?
|f38-i01-d05-searchspace| |f38-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{38}$ in dimension 5 for the first instance.\\[1em]
|f38-i01-d05-logobjspace| |f38-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 38 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{38}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f38-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f38-i01-d05-searchspace.*
:width: 49%
.. |f38-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f38-i01-d05-searchspace-projection.*
:width: 49%
.. |f38-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f38-i01-d05-logobjspace.*
:width: 49%
.. |f38-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f38-i01-d05-objspace.*
:width: 49%
.. _f39:
:math:`f_{39}`: Sharp ridge/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sharp ridge function (|f`13` in the bbob suite|_) and the
Schwefel function (|f`20` in the bbob suite|_).
While the first objective function is unimodal, non-separable, and
non-differentiable at its ridge, the second objective function is highly
multimodal---having the most prominent :math:`2^n` minima located
comparatively close to the corners of its unpenalized search area.
Contained in the *ill-conditioned - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the different difficulties (in particular
the non-differentiability of the first and the high multi-modality
and weak global structure of the second objective) when approaching
the Pareto front/Pareto set, especially in the middle?
|f39-i01-d05-searchspace| |f39-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{39}$ in dimension 5 for the first instance.\\[1em]
|f39-i01-d05-logobjspace| |f39-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 39 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{39}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f39-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f39-i01-d05-searchspace.*
:width: 49%
.. |f39-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f39-i01-d05-searchspace-projection.*
:width: 49%
.. |f39-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f39-i01-d05-logobjspace.*
:width: 49%
.. |f39-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f39-i01-d05-objspace.*
:width: 49%
.. _f40:
:math:`f_{40}`: Sharp ridge/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sharp ridge function (|f`13` in the bbob suite|_) and the
Gallagher function with 101 peaks (|f`21` in the bbob suite|_).
Both objective functions are non-separable.
While the first objective function is unimodal and non-differentiable at
its ridge, the second objective function
has 101 optima with position and height being unrelated and randomly
chosen (different for each instantiation of the function). The
conditioning around the global optimum of the second objective function
is about 30.
Contained in the *ill-conditioned - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Is the search effective without any global structure?
.. * How much does the multi-modality of the second objective play a role
when compared to fully uni-modal functions?
|f40-i01-d05-searchspace| |f40-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{40}$ in dimension 5 for the first instance.\\[1em]
|f40-i01-d05-logobjspace| |f40-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 40 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{40}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f40-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f40-i01-d05-searchspace.*
:width: 49%
.. |f40-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f40-i01-d05-searchspace-projection.*
:width: 49%
.. |f40-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f40-i01-d05-logobjspace.*
:width: 49%
.. |f40-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f40-i01-d05-objspace.*
:width: 49%
.. _f41:
:math:`f_{41}`: Sum of different powers/Sum of different powers
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two sum of different powers functions
(|f`14` in the bbob suite|_).
Both functions are uni-modal and non-separable where the sensitivies of
the variables in the rotated search space become more and more different
when approaching the objectives' optima.
Contained in the *ill-conditioned - ill-conditioned* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_{11}`: What is the effect of rotations
of the search space and missing self-similarity?
|f41-i01-d05-searchspace| |f41-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{41}$ in dimension 5 for the first instance.\\[1em]
|f41-i01-d05-logobjspace| |f41-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 41 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{41}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f41-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f41-i01-d05-searchspace.*
:width: 49%
.. |f41-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f41-i01-d05-searchspace-projection.*
:width: 49%
.. |f41-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f41-i01-d05-logobjspace.*
:width: 49%
.. |f41-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f41-i01-d05-objspace.*
:width: 49%
.. _f42:
:math:`f_{42}`: Sum of different powers/Rastrigin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sum of different powers functions
(|f`14` in the bbob suite|_) and the Rastrigin function
(|f`15` in the bbob suite|_).
Both objective functions are non-separable. While the first one
is unimodal, the second objective
function is highly multi-modal (roughly :math:`10^n` local optima).
Contained in the *ill-conditioned - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of having to cope with an increasing conditioning
when optimizing one objective function and the high multi-modality of the
other, in particular when trying to approximate the Pareto front/Pareto set
not close to their extremes?
|f42-i01-d05-searchspace| |f42-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{42}$ in dimension 5 for the first instance.\\[1em]
|f42-i01-d05-logobjspace| |f42-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 42 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{42}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f42-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f42-i01-d05-searchspace.*
:width: 49%
.. |f42-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f42-i01-d05-searchspace-projection.*
:width: 49%
.. |f42-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f42-i01-d05-logobjspace.*
:width: 49%
.. |f42-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f42-i01-d05-objspace.*
:width: 49%
.. _f43:
:math:`f_{43}`: Sum of different powers/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sum of different powers functions
(|f`14` in the bbob suite|_) and the Schaffer F7 function with
condition number 10 (|f`17` in the bbob suite|_).
Both objective functions are non-separable. While the first one
is unimodal with an increasing conditioning once the optimum is approached,
the second objective function is asymmetric and highly multi-modal with a
low conditioning where frequency and amplitude of the modulation vary.
Contained in the *ill-conditioned - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the different difficulties when approaching
the Pareto front/Pareto set, especially in the middle?
|f43-i01-d05-searchspace| |f43-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{43}$ in dimension 5 for the first instance.\\[1em]
|f43-i01-d05-logobjspace| |f43-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 43 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{43}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f43-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f43-i01-d05-searchspace.*
:width: 49%
.. |f43-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f43-i01-d05-searchspace-projection.*
:width: 49%
.. |f43-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f43-i01-d05-logobjspace.*
:width: 49%
.. |f43-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f43-i01-d05-objspace.*
:width: 49%
.. _f44:
:math:`f_{44}`: Sum of different powers/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sum of different powers functions
(|f`14` in the bbob suite|_) and the Schwefel function (|f`20` in the bbob suite|_).
Both objectives are non-separable.
While the first objective function is unimodal,
the second objective function is highly multimodal---having the most
prominent :math:`2^n` minima located comparatively close to the corners
of its unpenalized search area.
Contained in the *ill-conditioned - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the different difficulties (in particular
the increasing conditioning close to the first objective's optimum
and the high multi-modality and weak global structure of the second
objective) when approaching the Pareto front/Pareto set, especially in
the middle?
|f44-i01-d05-searchspace| |f44-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{44}$ in dimension 5 for the first instance.\\[1em]
|f44-i01-d05-logobjspace| |f44-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 44 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{44}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f44-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f44-i01-d05-searchspace.*
:width: 49%
.. |f44-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f44-i01-d05-searchspace-projection.*
:width: 49%
.. |f44-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f44-i01-d05-logobjspace.*
:width: 49%
.. |f44-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f44-i01-d05-objspace.*
:width: 49%
.. _f45:
:math:`f_{45}`: Sum of different powers/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the sum of different powers functions
(|f`14` in the bbob suite|_) and the Gallagher function with
101 peaks (|f`21` in the bbob suite|_).
Both objective functions are non-separable.
While the first objective function is unimodal, the second objective function
has 101 optima with position and height being unrelated and randomly
chosen (different for each instantiation of the function). The
conditioning around the global optimum of the second objective function
is about 30.
Contained in the *ill-conditioned - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Is the search effective without any global structure?
.. * How much does the multi-modality of the second objective play a role
when compared to fully uni-modal functions?
|f45-i01-d05-searchspace| |f45-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{45}$ in dimension 5 for the first instance.\\[1em]
|f45-i01-d05-logobjspace| |f45-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 45 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{45}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f45-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f45-i01-d05-searchspace.*
:width: 49%
.. |f45-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f45-i01-d05-searchspace-projection.*
:width: 49%
.. |f45-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f45-i01-d05-logobjspace.*
:width: 49%
.. |f45-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f45-i01-d05-objspace.*
:width: 49%
.. _f46:
:math:`f_{46}`: Rastrigin/Rastrigin
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two Rastrigin functions
(|f`15` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal
(roughly :math:`10^n` local optima).
Contained in the *multi-modal - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * When compared to :math:`f_{11}`: What is the effect of non-separability and
multi-modality?
|f46-i01-d05-searchspace| |f46-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{46}$ in dimension 5 for the first instance.\\[1em]
|f46-i01-d05-logobjspace| |f46-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 46 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{46}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f46-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f46-i01-d05-searchspace.*
:width: 49%
.. |f46-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f46-i01-d05-searchspace-projection.*
:width: 49%
.. |f46-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f46-i01-d05-logobjspace.*
:width: 49%
.. |f46-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f46-i01-d05-objspace.*
:width: 49%
.. _f47:
:math:`f_{47}`: Rastrigin/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rastrigin function
(|f`15` in the bbob suite|_) and the Schaffer F7 function with
condition number 10 (|f`17` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal.
Contained in the *multi-modal - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the different distributions of local minima
when approaching the Pareto front/Pareto set, especially in the middle?
|f47-i01-d05-searchspace| |f47-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{47}$ in dimension 5 for the first instance.\\[1em]
|f47-i01-d05-logobjspace| |f47-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 47 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{47}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f47-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f47-i01-d05-searchspace.*
:width: 49%
.. |f47-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f47-i01-d05-searchspace-projection.*
:width: 49%
.. |f47-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f47-i01-d05-logobjspace.*
:width: 49%
.. |f47-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f47-i01-d05-objspace.*
:width: 49%
.. _f48:
:math:`f_{48}`: Rastrigin/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rastrigin function
(|f`15` in the bbob suite|_) and the Schwefel function (|f`20` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal where
the first has roughly :math:`10^n` local optima and the most prominent
:math:`2^n` minima of the second objective function are located
comparatively close to the corners of its unpenalized search area.
Contained in the *multi-modal - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * What is the effect of the large amount of local optima in both objectives
when approaching the Pareto front/Pareto set, especially in the middle?
|f48-i01-d05-searchspace| |f48-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{48}$ in dimension 5 for the first instance.\\[1em]
|f48-i01-d05-logobjspace| |f48-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 48 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{48}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f48-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f48-i01-d05-searchspace.*
:width: 49%
.. |f48-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f48-i01-d05-searchspace-projection.*
:width: 49%
.. |f48-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f48-i01-d05-logobjspace.*
:width: 49%
.. |f48-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f48-i01-d05-objspace.*
:width: 49%
.. _f49:
:math:`f_{49}`: Rastrigin/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Rastrigin function
(|f`15` in the bbob suite|_) and the Gallagher function with
101 peaks (|f`21` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal where
the first has roughly :math:`10^n` local optima and the second has
101 optima with position and height being unrelated and randomly
chosen (different for each instantiation of the function).
Contained in the *multi-modal - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Is the search effective without any global structure?
.. * What is the effect of the differing distributions of local optima
in the two objective functions?
|f49-i01-d05-searchspace| |f49-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{49}$ in dimension 5 for the first instance.\\[1em]
|f49-i01-d05-logobjspace| |f49-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 49 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{49}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f49-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f49-i01-d05-searchspace.*
:width: 49%
.. |f49-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f49-i01-d05-searchspace-projection.*
:width: 49%
.. |f49-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f49-i01-d05-logobjspace.*
:width: 49%
.. |f49-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f49-i01-d05-objspace.*
:width: 49%
.. _f50:
:math:`f_{50}`: Schaffer F7, condition 10/Schaffer F7, condition 10
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two Schaffer F7 functions with
condition number 10 (|f`17` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal.
Contained in the *multi-modal - multi-modal* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_{46}`: What is the effect of multimodality
on a less regular function?
|f50-i01-d05-searchspace| |f50-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{50}$ in dimension 5 for the first instance.\\[1em]
|f50-i01-d05-logobjspace| |f50-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 50 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{50}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f50-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f50-i01-d05-searchspace.*
:width: 49%
.. |f50-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f50-i01-d05-searchspace-projection.*
:width: 49%
.. |f50-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f50-i01-d05-logobjspace.*
:width: 49%
.. |f50-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f50-i01-d05-objspace.*
:width: 49%
.. _f51:
:math:`f_{51}`: Schaffer F7, condition 10/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Schaffer F7 function with
condition number 10 (|f`17` in the bbob suite|_)
and the Schwefel function (|f`20` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal.
While frequency and amplitude of the modulation vary in an almost
regular fashion in the first objective function, the second objective
function posseses less global structure.
Contained in the *multi-modal - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * What are the effects of different global structures in the two
objective functions?
|f51-i01-d05-searchspace| |f51-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{51}$ in dimension 5 for the first instance.\\[1em]
|f51-i01-d05-logobjspace| |f51-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 51 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{51}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f51-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f51-i01-d05-searchspace.*
:width: 49%
.. |f51-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f51-i01-d05-searchspace-projection.*
:width: 49%
.. |f51-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f51-i01-d05-logobjspace.*
:width: 49%
.. |f51-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f51-i01-d05-objspace.*
:width: 49%
.. _f52:
:math:`f_{52}`: Schaffer F7, condition 10/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Schaffer F7 function with
condition number 10 (|f`17` in the bbob suite|_)
and the Gallagher function with
101 peaks (|f`21` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal.
While frequency and amplitude of the modulation vary in an almost
regular fashion in the first objective function, the second has
101 optima with position and height being unrelated and randomly
chosen (different for each instantiation of the function).
Contained in the *multi-modal - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Similar to :math:`f_{51}`: What are the effects of different
global structures in the two objective functions?
|f52-i01-d05-searchspace| |f52-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{52}$ in dimension 5 for the first instance.\\[1em]
|f52-i01-d05-logobjspace| |f52-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 52 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{52}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f52-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f52-i01-d05-searchspace.*
:width: 49%
.. |f52-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f52-i01-d05-searchspace-projection.*
:width: 49%
.. |f52-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f52-i01-d05-logobjspace.*
:width: 49%
.. |f52-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f52-i01-d05-objspace.*
:width: 49%
.. _f53:
:math:`f_{53}`: Schwefel x*sin(x)/Schwefel x*sin(x)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two Schwefel functions (|f`20` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal where
the most prominent :math:`2^n` minima of each objective function are
located comparatively close to the corners of its unpenalized search area.
Due to the combinatorial nature of the Schwefel function, it is likely
in low dimensions that the Pareto set goes through the origin of the
search space.
Contained in the *weakly-structured - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison with :math:`f_{50}`: What is the effect of a weak global
structure?
.. * Can the search algorithm benefit from Pareto-optimal search points
it can get from random samples close to the origin on some of the
function' instances?
|f53-i01-d05-searchspace| |f53-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{53}$ in dimension 5 for the first instance.\\[1em]
|f53-i01-d05-logobjspace| |f53-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 53 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{53}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f53-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f53-i01-d05-searchspace.*
:width: 49%
.. |f53-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f53-i01-d05-searchspace-projection.*
:width: 49%
.. |f53-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f53-i01-d05-logobjspace.*
:width: 49%
.. |f53-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f53-i01-d05-objspace.*
:width: 49%
.. _f54:
:math:`f_{54}`: Schwefel x*sin(x)/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of the Schwefel function (|f`20` in the bbob suite|_) and the Gallagher function with
101 peaks (|f`21` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal.
For the first objective function, the most prominent :math:`2^n` minima
are located comparatively close to the corners of its unpenalized search
area. For the second objective, position and height of all
101 optima are unrelated and randomly
chosen (different for each instantiation of the function).
Contained in the *weakly-structured - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * In comparison to :math:`f_{53}`: Does the total absence of a global
structure in one objective change anything in the performance of the
algorithm?
|f54-i01-d05-searchspace| |f54-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{54}$ in dimension 5 for the first instance.\\[1em]
|f54-i01-d05-logobjspace| |f54-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 54 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{54}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f54-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f54-i01-d05-searchspace.*
:width: 49%
.. |f54-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f54-i01-d05-searchspace-projection.*
:width: 49%
.. |f54-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f54-i01-d05-logobjspace.*
:width: 49%
.. |f54-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f54-i01-d05-objspace.*
:width: 49%
.. _f55:
:math:`f_{55}`: Gallagher 101 peaks/Gallagher 101 peaks
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Combination of two Gallagher functions with
101 peaks (|f`21` in the bbob suite|_).
Both objective functions are non-separable and highly multi-modal.
Position and height of all 101 optima in each objective function
are unrelated and randomly chosen and thus, no global structure
is present.
Contained in the *weakly-structured - weakly-structured* function class.
.. .. rubric:: Information gained from this function:
.. * Can the Pareto front/Pareto set be found efficiently when no global
structure can be exploited?
|f55-i01-d05-searchspace| |f55-i01-d05-searchspace-projected|
.. raw:: latex
Illustration of search space for \code{bbob-biobj} function $f_{55}$ in dimension 5 for the first instance.\\[1em]
|f55-i01-d05-logobjspace| |f55-i01-d05-objspace|
.. raw:: html
Illustration of search space (first row) and objective space (second row) for
bbob-biobj
function 55 in dimension 5 for the first instance.
.. raw:: latex
Illustration of objective space for \code{bbob-biobj} function $f_{55}$ in dimension 5 for the
first instance (left: normalized in log-scale; right: original scaling). \pagebreak
.. |f55-i01-d05-searchspace| image:: ../code/plots/after_workshop/directions-f55-i01-d05-searchspace.*
:width: 49%
.. |f55-i01-d05-searchspace-projected| image:: ../code/plots/after_workshop/directions-f55-i01-d05-searchspace-projection.*
:width: 49%
.. |f55-i01-d05-logobjspace| image:: ../code/plots/after_workshop/directions-f55-i01-d05-logobjspace.*
:width: 49%
.. |f55-i01-d05-objspace| image:: ../code/plots/after_workshop/directions-f55-i01-d05-objspace.*
:width: 49%
The Extended ``bbob-biobj-ext`` Test Suite and Its Functions
============================================================
Having all combinations of only a subset of the single-objective ``bbob`` functions in a test suite
like the above ``bbob-biobj`` one has
advantages but also a few disadvantages. Using only a subet of the 24 ``bbob`` functions
introduces a bias towards the chosen functions and reduces the amount of different difficulties,
a bi-objective algorithm is exposed to in the benchmarking exercise. Allowing all combinations of
``bbob`` functions increases the percentage of problems for which both objectives are from different
``bbob`` function groups while, in practice, it can often be assumed that both objective functions
come from a similar "function domain".
The rationale behind the following extended ``bbob-biobj`` test suite, denoted as ``bbob-biobj-ext``,
is to reduce the mentioned effects. To this end, we add all within-group combinations of ``bbob``
functions which are not already in the ``bbob-biobj`` suite and which do not combine a function
with itself. For technical reasons, we also remove the Weierstrass functions (|fb16|_ in the
``bbob`` suite) because the optimum is not necessarily unique and computing the nadir point
therefore technically more challenging than for the other functions.
This extension adds :math:`3*(4+3+2+1-1) + 2*(3+2+1-1) = 3*9+2*5=37` functions, resulting in
92 functions overall.
The following table details which single-objective ``bbob`` functions are contained in the
92 ``bbob-biobj-ext`` functions (outer column and row annotations) and indicates their IDs.
Note that the IDs of the first 55 ``bbob0biobj-ext`` functions are the same than for the
``bbob-biobj`` test suite for compatibility reasons.
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
| ||fb1|_ ||fb2|_ ||fb3|_ ||fb4|_ ||fb5|_ ||fb6|_ ||fb7|_ ||fb8|_ ||fb9|_ ||fb10|_||fb11|_||fb12|_||fb13|_||fb14|_||fb15|_||fb16|_||fb17|_||fb18|_||fb19|_||fb20|_||fb21|_||fb22|_||fb23|_||fb24|_|
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb1|_ | |f1| | |f2| | |f56| | |f57| | |f58| | |f3| | | |f4| | | | | | |f5| | |f6| | |f7| | | |f8| | | | |f9| | |f10| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb2|_ | | |f11| | |f59| | |f60| | |f61| | |f12| | | |f13| | | | | | |f14| | |f15| | |f16| | | |f17| | | | |f18| | |f19| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb3|_ | | | | |f62| | |f63| | | | | | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb4|_ | | | | | |f64| | | | | | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb5|_ | | | | | | | | | | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb6|_ | | | | | | |f20| | |f65| | |f21| | |f66| | | | | |f22| | |f23| | |f24| | | |f25| | | | |f26| | |f27| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb7|_ | | | | | | | | |f67| | |f68| | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb8|_ | | | | | | | | |f28| | |f69| | | | | |f29| | |f30| | |f31| | | |f32| | | | |f33| | |f34| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb9|_ | | | | | | | | | | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb10|_| | | | | | | | | | | |f70| | |f71| | |f72| | |f73| | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb11|_| | | | | | | | | | | | |f74| | |f75| | |f76| | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb12|_| | | | | | | | | | | | | |f77| | |f78| | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb13|_| | | | | | | | | | | | | |f35| | |f36| | |f37| | | |f38| | | | |f39| | |f40| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb14|_| | | | | | | | | | | | | | |f41| | |f42| | | |f43| | | | |f44| | |f45| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb15|_| | | | | | | | | | | | | | | |f46| | | |f47| | |f79| | |f80| | |f48| | |f49| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb16|_| | | | | | | | | | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb17|_| | | | | | | | | | | | | | | | | |f50| | |f81| | |f82| | |f51| | |f52| | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb18|_| | | | | | | | | | | | | | | | | | | |f83| | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb19|_| | | | | | | | | | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb20|_| | | | | | | | | | | | | | | | | | | | |f53| | |f54| | |f84| | |f85| | |f86| |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb21|_| | | | | | | | | | | | | | | | | | | | | |f55| | |f87| | |f88| | |f89| |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb22|_| | | | | | | | | | | | | | | | | | | | | | | |f90| | |f91| |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb23|_| | | | | | | | | | | | | | | | | | | | | | | | |f92| |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
||fb24|_| | | | | | | | | | | | | | | | | | | | | | | | |
+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+
The 92 functions of the ``bbob-biobj-ext`` test suite and their IDs (in the table cells) together with the information about which single-objective ``bbob`` functions are used
to define them (outer column and row annotations).
.. |fb3| replace:: :math:`f_3`
.. _fb3: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=15
.. |fb4| replace:: :math:`f_4`
.. _fb4: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=20
.. |fb5| replace:: :math:`f_5`
.. _fb5: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=25
.. |fb7| replace:: :math:`f_7`
.. _fb7: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=35
.. |fb9| replace:: :math:`f_9`
.. _fb9: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=45
.. |fb10| replace:: :math:`f_{10}`
.. _fb10: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=50
.. |fb11| replace:: :math:`f_{11}`
.. _fb11: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=55
.. |fb12| replace:: :math:`f_{12}`
.. _fb12: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=60
.. |fb16| replace:: :math:`f_{16}`
.. _fb16: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=80
.. |fb18| replace:: :math:`f_{18}`
.. _fb18: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=90
.. |fb19| replace:: :math:`f_{19}`
.. _fb19: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=95
.. |fb22| replace:: :math:`f_{22}`
.. _fb22: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=110
.. |fb23| replace:: :math:`f_{23}`
.. _fb23: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=115
.. |fb24| replace:: :math:`f_{24}`
.. _fb24: http://coco.lri.fr/downloads/download15.03/bbobdocfunctions.pdf#page=120
.. |f56| replace:: f56
.. |f57| replace:: f57
.. |f58| replace:: f58
.. |f59| replace:: f59
.. |f60| replace:: f60
.. |f61| replace:: f61
.. |f62| replace:: f62
.. |f63| replace:: f63
.. |f64| replace:: f64
.. |f65| replace:: f65
.. |f66| replace:: f66
.. |f67| replace:: f67
.. |f68| replace:: f68
.. |f69| replace:: f69
.. |f70| replace:: f70
.. |f71| replace:: f71
.. |f72| replace:: f72
.. |f73| replace:: f73
.. |f74| replace:: f74
.. |f75| replace:: f75
.. |f76| replace:: f76
.. |f77| replace:: f77
.. |f78| replace:: f78
.. |f79| replace:: f79
.. |f80| replace:: f80
.. |f81| replace:: f81
.. |f82| replace:: f82
.. |f83| replace:: f83
.. |f84| replace:: f84
.. |f85| replace:: f85
.. |f86| replace:: f86
.. |f87| replace:: f87
.. |f88| replace:: f88
.. |f89| replace:: f89
.. |f90| replace:: f90
.. |f91| replace:: f91
.. |f92| replace:: f92
Function Groups
---------------------------------------------------------------
Like for the ``bbob-biobj`` test suite, we obtain 15 function
classes to structure the 92 bi-objective functions of the ``bbob-biobj-ext`` test
suite. Depending on whether a function class combines functions from the same or
from different ``bbob`` function classes, each function class contains
8, 12 or just four functions. We are listing
below the function classes and in parenthesis the functions that belong to
the respective class:
1. separable - separable (12 functions: f1, f2, f11, f56-64)
2. separable - moderate (f3, f4, f12, f13)
3. separable - ill-conditioned (f5, f6, f14, f15)
4. separable - multi-modal (f7, f8, f16, f17)
5. separable - weakly-structured (f9, f10, f18, f19)
6. moderate - moderate (8 functions: f20, f21, f28, f65-f69)
7. moderate - ill-conditioned (f22, f23, f29, f30)
8. moderate - multi-modal (f24, f25, f31, f32)
9. moderate - weakly-structured (f26, f27, f33, f34)
10. ill-conditioned - ill-conditioned (12 functions: f35, f36, f41, f70-78)
11. ill-conditioned - multi-modal (f37, f38, f42, f43)
12. ill-conditioned - weakly-structured (f39, f40, f44, f45)
13. multi-modal - multi-modal (8 functions: f46, f47, f50, f79-83)
14. multi-modal - weakly structured (f48, f49, f51, f52)
15. weakly structured - weakly structured (12 functions: f53-55, f84-92)
Normalization and Instances
---------------------------
Normalization of the objectives and instances are handled for the ``bbob-biobj-ext`` in the
same manner as for the ``bbob-biobj`` suite, i.e., no normalization of the objective
functions is taking place for the algorithm benchmarking and 15 instances are prescribed for
a typical experiment.
.. _`Coco framework`: https://github.com/numbbo/coco
.. raw:: html