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The CEC benchmark environments do not establish subgroups of constrained problems. That is, results obtained by an algorithm can hardly be identified with a certain problem characteristic. Although, the CEC2017 collection would allow for a rough categorizations. For example, the constrained problems [MATH] [MATH] , and... |
All CEC benchmark sets share the definition of a feasible solution introduced in Sec. Due to the issue of enforcing the generation of candidate solutions that exactly satisfy the equality constraints, the error margin of [MATH] is used in all three competitions. |
Every algorithm has to perform [MATH] independent runs on a single instance of each constrained optimization problem. In each run, the best result so far [MATH] is monitored at three distinct points of the search process, i.e. after 10%, after 50%, and after 100% of the assigned function evaluation budget have been con... |
To this end, an algorithm is required to report the best so far objective function value [MATH] , the corresponding mean constraint violation [MATH] , as well as the triplet [MATH] (see Table ). The mean constraint violation [MATH] of a candidate solution [MATH] is determined as |
[EQUATION] where [MATH] is the aggregated number of equality and inequality constraints of problem ( COP ). Note that the constraint violation [MATH] is obtained according to Eq. ( ). The term [MATH] specifies the number of violated constraints with violation greater than [MATH] [MATH] , and [MATH] , respectively. |
The results of these [MATH] runs are then used to compute statistics for algorithm evaluation and comparison. In order to sort the realized candidate solutions, the CEC benchmarks introduce a lexicographic ordering with respect to [MATH] and [MATH] That is, two candidate solutions [MATH] and [MATH] are sorted according... |
[EQUATION] Note that Eq. ( 10 ) is defined analogously to the order relation ( ), but makes use of Eq. ( ) instead of Eq. ( ). A comprehensive list of the utilized quality indicators is provided in Table |
The CEC2006 benchmark set provided the globally optimal parameter vectors of each test problem. Using this information the effectiveness of an algorithm was determined in terms of the deviation [MATH] of the best-so-far solution [MATH] from the optimum [MATH] It was further used to calculate the success rate ( [MATH] )... |
[EQUATION] is realized. Note that, by distinguishing two feasible candidate solutions based on their deviation from the known optimum, the CEC2006 benchmarks use a slightly different way of proceeding than presented in ( 10 ). |
No longer having information about the global optima, the success rate was replaced with the calculation of the feasibility rate ( [MATH] ) in the succeeding CEC competitions. [MATH] indicates the ratio of those algorithm runs that realized at least one feasible solution and the total number of algorithm runs. |
Regarding the termination criterion used by the CEC competitions, each constrained problem comes with a fixed budget of function evaluations. |
Termination is required after an algorithm has entirely consumed this budget. The budget of function evaluations allocated to each constrained function varies among competitions. While it is fixed to [MATH] function evaluations (regardless of the problem dimension) for CEC2006, the CEC2010 and CEC2017 collections defin... |
4.3 Reporting By primarily representing test problems for the CEC competitions on constrained real-parameter optimization, the corresponding technical reports do not make a statement on ensuring newsworthiness of the algorithm results. In order to participate in the mentioned competitions, algorithm results have to be ... |
in the context of unconstrained benchmarks, the comparison of novel algorithmic ideas with diverse state-of-the-art strategies is essential to prevent the publication of already dominated results and to contribute to real progress in the respective field of research. |
The final quality indicators computed for a specific algorithm have to be presented for every single constrained problem in a detailed table. Considering that the CEC benchmarks demand information on three stages ( [MATH] [MATH] , and [MATH] ) of the search process, this presentation style appears rather lengthy. Table... |
Making use of one table per dimension, and per algorithm, leads to increasing space requirements when considering more search space dimensions. Furthermore, drawing conclusions with respect to algorithm performance differences is made very difficult. Additionally, not subsuming problems of similar characteristics imped... |
The CEC2006 and CEC2010 benchmarks were using convergence graphs to provide a more tangible notion of algorithm performance. In 2006, the convergence graphs illustrated the deviation of the objective function value from the optimum [MATH] as well as the mean constraint violation [MATH] plotted against the number of fun... |
The benchmark collections demand to report the configuration of the PC on which the experiments have been executed. To this end, the operation system , the CPU , the memory , the programming language used, and the algorithm have to be specified. Acting this way intends to support algorithm comparability. However, a per... |
With respect to algorithm reporting, the CEC related technical reports require the complete description of the algorithm parameters used as well as their specific ranges. Further, algorithm designers are demanded to present guidelines for potential parameter adjustments and estimates of the corresponding costs in terms... |
In order to give an impression of the algorithm complexity (see Sec. 3.3 ), three quantities have to be presented. The average [MATH] of the computation time [MATH] of [MATH] evaluations, as well as [MATH] , the complete computation time [MATH] |
of a specific algorithm over all problems [MATH] of similar dimensionality [EQUATION] Here, [MATH] denotes the number of constrained optimization problems with similar dimensionality of a respective benchmark function set. |
[MATH] and [MATH] are reported together with their relative difference [MATH] To represent a meaningful quantity of algorithm complexity [MATH] the measurements [MATH] and [MATH] need to consider a sufficiently large number of function evaluations. However, such an approach can be problematic: Imagine a DE algorithm |
(or an EDA ), that initializes a rather large archive of about [MATH] candidate solutions. Considering dimension [MATH] , such an algorithm would consume the whole budget of [MATH] function evaluations in its initialization process. Consequently, [MATH] cannot provide any information about the actual algorithm running ... |
Considering the ranking of competing algorithms, the presentation style promotes the need for a well-defined algorithm ranking. Unfortunately, the technical report of the CEC2006 competition |
does not provide any motivation of a suitable ranking procedure at all. The presentation of the competition results is also of little help. Hence, the quality indicators used to obtain an algorithm ranking cannot be deduced. |
While defined in different ways, the ranking schemes used for the CEC2010 and CEC2017 benchmarks are fully explained. The CEC2010 ranking method is based on a mean value comparison of two or more algorithms on each individual constrained problem. Algorithms that yield feasibility rates of [MATH] are ordered based on th... |
[EQUATION] This way the best algorithm is defined by the lowest rank value [MATH] The CEC2017 ranking method is considering the mean objective function values as well as the median solution at the maximal allowed number of function evaluations. The first ranking of all competing algorithms is based on the mean values. ... |
[EQUATION] Again, the best algorithm obtains the lowest rank value [MATH] Regarding these two ranking methods, it is noticed that the CEC2017 ranking is a progression. It no longer uses a single ranking (average case quality in the broadest sense), but the consensus of average case and median case quality. This is in l... |
, where the use of so-called consensus rankings is recommended for algorithm comparison. Consensus rankings are distinguished into positional and optimization-based methods. |
The definition of a consensus ranking is by no means unique as it is rather sensitive with respect to the choice of individual rankings and the number of considered algorithms. A desirable property of a consensus ranking would be the Independence of Irrelevant Alternatives (IIA) criterion |
stating that changes in the number of algorithms must not affect the pair-wise preference in the consensus ranks. That is, if the consensus ranks algorithm [MATH] first and algorithm [MATH] second among five distinct algorithms, then disregarding any other algorithm should not yield a consensus rank change between [MAT... |
The positional consensus ranking of the CEC2017 benchmarks is created by simply adding the mean and median ranks. This can result in potentially undesirable consensus rankings . For example, consider the scenario of comparing three distinct algorithms A1, A2, A3 with mean ranking [MATH] and median ranking [MATH] on a s... |
Moreover, the CEC ranking approaches aggregate algorithm rankings over multiple dimensions. This way, algorithms which are especially well performing in lower dimensions are potentially overrated and the overall ranking might be prejudiced. Further, algorithms that are particularly well performing in larger dimensions ... |
To conclude this review of the constrained CEC benchmarks, some of the mentioned aspects could be incorporated in the advancing CEC competitions on constrained real-parameter optimization. In doing so, algorithm developers would benefit from the introduction of well-designed problem subgroups that support the identific... |
Table recaps the vital benefits of the CEC benchmarks as well as some room for improvement which has been mentioned in more detail within this and the previous subsections. However, not all of the mentioned improvements can automatically be considered a shortcoming of the CEC benchmarks. The benchmarks might rather be ... |
The COCO framework The Comparing Continuous Optimizer (COCO) suite provides a platform to benchmark and compare continuous optimizers for numerical (non-linear) optimization. Only recently, the development of a COCO branch for constrained optimization problems started. The related code is available on the project websi... |
While the COCO BBOB-constrained testbed is not yet operational, being short before completion, the corresponding benchmarking principles and the associated test problem structure are not expected to substantially change anymore. As the COCO framework represents the currently most elaborated benchmarking environment for... |
However, caution is advised with respect to small changes in individual test function aspects, e.g. the distances of the constrained optimal solution from the unconstrained optimal solution or regarding the post-processing practice. |
The rest of this section is concerned with pointing out the COCO BBOB-constrained benchmarking conventions, the related test problem definitions, the evaluation criteria as well as the presentation style. |
5.1 Benchmarking principles The COCO BBOB-constrained suite is distinctly built on the unconstrained COCO framework. The COCO platform assists algorithm engineers in setting up proper experiments for algorithm comparison. It provides simple interfaces to multiple programming languages (C/C++, Python, MATLAB/Octave, and... |
The benchmark functions are considered to represent black-box functions for the tested algorithms. Still, the objective functions are explicitly stated in mathematical form in the documentation. This allows for a deeper understanding of the individual problem difficulties and thus of an algorithm’s (in)capabilities. In... |
The COCO guideline for counting function evaluations in the constrained setting involves distinguishing objective function evaluations and constraint evaluations. Still, one constraint evaluation is identified with the evaluation of all individual constraint functions at a time. Accordingly, a specified budget of funct... |
On the one hand, the formal constrained function definitions are not specifying any box-constraints, refer to ( 15 ). In this regard, guidelines for the treatment of box-constraints are not needed. Yet, the BBOB-constrained suite provides the user with the subroutines cocoProblemGetSmallestValuesOfInterest , and cocoPr... |
Whether the box-constraints need to be enforced in every step or not is of course a design question. Anyway, the benchmark designers need to provide plain instructions with respect to treatment of box-constraints during the search process. The use of the box-constraint handling may be beneficial on some constrained pro... |
5.2 Experimental design The standard BBOB-constrained optimization problem reads [EQUATION] A summary of the associated problem features is provided in Table The considered constrained functions are separated into eight subgroups associated with the selected objective functions. These objective functions are |
1. the Sphere function, 2. the Ellipsoid function, 3. the Linear slope function, 4. the rotated Ellipsoid function, 5. the rotated Discuss function, |
6. the rotated Bent Cigar, 7. the rotated Different Powers, and 8. the rotated Rastrigin function. By systematically equipping each objective function with 6 different numbers of inequality constraint functions, namely [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] constraints, the BBOB-constrained benchmark problems ... |
For now, the COCO BBOB-constrained testbed concentrates on almost linear inequality constraints. To this end, the linear structure of the feasible region is distorted by application of bijective non-linear transformations on a number of constrained functions. The subsequent application of a randomly generated translati... |
The procedure to create a constrained function consists of five steps I. Select a pseudo-convex objective function [MATH] and a corresponding number [MATH] of constraints [MATH] |
II. Define the first linear constraint [MATH] III. Construct the remaining linear constraints [MATH] by sampling their gradients from a multivariate normal distribution and incrementally demanding that the origin remains a Karush-Kuhn-Tucker (KKT) point of the problem |
IV. If applicable, apply non-linear transformations to the constrained function. V. Randomly sample a translation vector to change the location of the optimal solution. |
According to the COCO BBOB-constrained documentation , the domain of almost linear constrained functions represents the most interesting starting configuration for benchmarking. Such constraint functions are composed of small variations of linear constraints which are considered to represent most simple restrictions to... |
The transformations are essentially applied to ensure constrained functions that are reasonably difficult to solve, i.e. potential regularities that might favor the exploitation abilities of certain algorithms are excluded. The transformations are designed in such a way that the automatic generation of similarly hard t... |
The constrained problems ( 15 ) are scalable with respect to search space dimension [MATH] and number of constrained functions [MATH] Taking into account dimensionality, objective function and the number of constraints, the BBOB-constrained testbed consists of [MATH] distinct constrained functions. By composing problem... |
When estimating the size of the feasible region relative to the box defined by the lower and upper bounds, the associated [MATH] values indicate the dependence of the dimension [MATH] Yet, the aggregated [MATH] value presented in Table only has limited significance. On the one hand, it was generated according to |
by considering only a single instance of each constrained function. As the randomly generated boundary of the feasible region may vary among constrained problem instances, the [MATH] value is supposed to exhibit fluctuations of some degree. On the other hand, the [MATH] was averaged over all possible problem dimensions... |
The benchmark suite does not determine a fixed budget of function evaluations. The specification of appropriate termination conditions for an individual algorithm is left to the user |
. In this context, the COCO built-in function cocoProblemFinalTargetHit delivers an indicator of the realized algorithm precision. It returns true after the algorithm has approached the optimal objective function value with accuracy [MATH] and can be utilized to terminate the algorithm run. Accordingly, the value of [M... |
By default, each algorithm is executed on [MATH] randomly generated instances of each constrained function. The corresponding results are interpreted as 15 independent repetitions on the same constrained problem. Acting this way prevents unintentional exploitation of potentially biasing function features |
Remember that the optimal solution is by construction located on the boundary of the feasible region. This property might potentially prejudice search algorithms to largely operate outside of the feasible region of the search space. Depending on the fitness environment, this allows for faster progress until the algorit... |
5.3 Reporting The COCO framework comes with a post-processing module for automated data preparation and visualization in terms of html or LaTeX templates. The user-independent standardization of the data processing reduces the susceptibility to errors and supports the comparability of algorithm performance. |
The COCO BBOB-constrained suite takes into account a single performance measure: the algorithm runtime. Runtime is defined in terms of the number of function evaluations consumed on a specific constrained problem until a predefined target is reached. In total, [MATH] targets uniformly distributed on the log-scale are s... |
This method permits to compare algorithms with different success rates. A detailed description is available in Whether a target was reached after evaluation of a candidate solution is automatically checked by the COCO suite. To this end, a trigger value is compared with the next unmatched target. The corresponding numb... |
Making use of this runtime definition results in a performance measure that is essentially independent of the computational platform and the programming language used. Further, the algorithm results can easily be condensed and presented in multiple ways, e.g. by measuring the average runtime (aRT) of an algorithm |
, by use of data profiles or empirical cumulative distribution function (ECDF) plots , or runtime tables for specific target values. An illustration of an aRT plot is displayed in Figure (a). It provides an estimate of the expected runtime. The aRT is computed by summing up all evaluations in unsuccessful algorithm run... |
Algorithms can be directly compared by illustrating their ECDFs per function evaluations in log-scales. This way, the area above and in between the graphs becomes a meaningful conception. An exemplary ECDF is illustrated in Figure . It can be interpreted in two ways: By considering the number of function evaluations on... |
Consequently, better performing algorithms realize smaller areas above a curve. Further, the difference between those areas can be interpreted as a measure of the performance advantage of one algorithm over another. With the caveat of loosing the connection to a single constrained problem, the ECDF plots allow for aggr... |
That is, the presentation of algorithm performance on problem subgroups is straight forward. Hence, in contrast to extensive and hardly interpretable tables, the ECDFs provide a relevant notion of algorithm suitability for single constrained functions, and subgroups of constrained problems, respectively. |
Only considering feasible candidate solutions in the trigger/target definition may inflate the relevance of late phases in the search process. Depending on the constrained problem, algorithms that sample an initial population within the box-constraints might consume a considerable number of function evaluations until t... |
Other trigger definitions are conceivable, i.e. the trigger may be defined by the sum of the objective function value and the constraint violation of a candidate solution. This way of proceeding takes into account infeasible steps, but it would introduce the issue of unwanted cancellation effects. Another idea to give ... |
The COCO experiments include the approximate measurement of the algorithm time complexity . To this end, it is recommended to monitor either the wall-clock or the CPU time while running the algorithm on the benchmark suite. The time normalized by the number of function evaluations is demanded to be reported for each di... |
As the development of the BBOB-constrained benchmark suite is still ongoing, the definitive presentation style of the algorithm results cannot be provided at this point. The presentation of additional information on the ratio of the feasible region relative to the box [MATH] is conceivable. Further, the ultimate choice... |
To wrap up the deliberations of Sec. , Table provides a shorthand overview of the benefits, as well as potential areas for improvement, of the BBOB-constrained testbed. It has to be noted that the mentioned improvements can not necessarily be regarded as a shortcoming of the respective benchmarking environment, as they... |
Conclusion The present review intends to collect principles for comparing constrained test environments for Evolutionary Algorithms. To this end, it takes into account recommendations on the basic principles, the experimental design, and the presentation of algorithm results. Based on the gathered criteria, the most pr... |
Both considered benchmark suites focus on different constrained problem domains. They differ in terms of counting function evaluations, defining termination criteria as well as performance evaluations comparison. The COCO BBOB-constrained benchmark is very much based on the unconstrained COCO framework. By including ex... |
These distinct benchmarking approaches directly induce different ways of presentation. On the one hand, the COCO framework measures runtime in terms of function evaluations per predefined target and visualizes algorithmic performance in terms of ECDF graphs. Algorithm performance can thus be rather easily aggregated ov... |
Both, the CEC competitions for constraint real-parameter optimization, and the COCO BBOB-constrained framework do only consider relaxable equality and inequality constraints. That is, the algorithms are allowed to move in the whole unconstrained search space. Each candidate solution, either feasible or infeasible, may ... |
Furthermore, both benchmark sets omit to demand a specific box-constraint treatment. Yet, they refrain from mentioning the need of the precise reporting of such approaches. Considering the source codes of the most successful strategies reported in CEC competitions, all algorithms were assuming situation (S1). Even if n... |
Considering the CEC benchmark environments, the problem definitions were subject to considerable changes in recent years. The introduction of scalable constrained functions was accompanied by a reduction of the average number of constraints per problem (from [MATH] to about [MATH] ). While the CEC2006 benchmarks were (... |
The constrained CEC benchmarks provide the currently most elaborated benchmarking environment for EA. They mainly present non-linearly constrained problems with a fixed number of not necessarily linear inequality and equality constraints. Also due to unconnected feasible sets, the CEC constrained test functions are con... |
Looking at the recent CEC2017 benchmarking functions, the distinct problem features introduce a rather high level of problem complexity. The benchmarks are suited to demonstrate the use of algorithmic ideas. According to its intention in the context of the CEC competition, the CEC benchmarking environment is well desig... |
Regardless of minor software bugs and unfinished post-processing methodology, the COCO BBOB-constrained suite could have the potential to become another standard constrained benchmarking platform. It is equipped with a detailed documentation of its benchmarking principles as well as an elaborated post-processing strate... |
However, comprising only a somehow limited number of distinct constrained problem types, algorithms that perform well on the COCO BBOB-constrained suite are not guaranteed to be successful on other constrained problems. In this respect, BBOB-constrained needs to proceed towards more complex constraint definitions of di... |
The COCO BBOB-constrained problem definitions might further permit (limited) user customizations. For instance, it should be possible to optionally move the optimal solution from the boundary into the feasible region. This could potentially increase the problem complexity for some constrained functions. Another option ... |
Taking into account the vast number of constrained problem characteristics, the current benchmarking environments under review do only cover a small ratio of the constrained problem domain. Both benchmarking environments might be extended with additional constrained test functions. However, a drawback of extending the ... |
Aiming at the establishment of profound benchmarks for real-valued constrained optimization, the two approaches should not be regarded as opposing but rather as complementing benchmarking suites. Both environments are based on reasonable design decisions that might need an upgrade but cannot be fully negated. Ultimatel... |
In the end, benchmarking environments have to demand diligent scientific investigations. In particular, algorithm developers must be urged to maintain reproducible and comparable algorithm results. Collecting principles for elaborate constrained benchmarking, Section can be regarded as a guideline for design convention... |
Advancing the CEC benchmark definitions, and finishing the COCO BBOB-constrained benchmark suite, are anticipated tasks for future research. Further the design of additional EA benchmarking tools for different constrained problem sub-domains needs to be challenged. A possible step in this direction might be the conside... |
, a unit hypercube of variable dimension with perturbed vertices, which represents the feasible region of the linear problem. The linear objective function is constructed in such a way that the Simplex algorithm yields an exponential worst-case running time. Considering the number of sophisticated deterministic approac... |
show poor performance in this environment. The Klee-Minty problem was already used to compare a specially designed CMSA-ES variant for linear optimization with open source interior point LP solvers in |
In case that this review fosters the impression of an unbalanced criticism, this conjecture is probably due to the fact that the constrained CEC benchmarks have existed for many years providing a multitude of benchmarking papers and working points, respectively. In contrast, there are hardly any algorithm comparisons t... |
# Source: arxiv 1806.04619 # Title: Isoperimetric inequalities in Riemann surfaces and graphs # Sections: all # Downloaded: 2026-03-03T02:23:03.584210+00:00 |
Isoperimetric inequalities in Riemann surfaces and graphs Abstract. A celebrated theorem of Kanai states that quasi-isometries preserve isoperimetric inequalities between uniform Riemannian manifolds (with positive injectivity radius) and graphs. Our main result states that we can study the (Cheeger) isoperimetric ineq... |
(1) Supported in part by a grant from Ministerio de Economía y Competitividad (MTM2015-63612P), Spain. (2) Supported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT),... |
Keywords: Isoperimetric inequality; Cheeger isoperimetric constant; Riemann surface; Poincaré metric; Gromov hyperbolicity. 2010 AMS Subject Classification numbers: Primary 53C21, 53C23; Secondary 58C40. |
1. Introduction Isoperimetric inequalities are of interest in pure and applied mathematics (see, e.g., ). There are close connections between isoperimetric inequality and some conformal invariants of Riemannian manifolds and graphs, namely Poincaré-Sobolev inequalities, the bottom of the spectrum of the Laplace-Beltram... |
10 , p.228] 40 , p.333] ). The Cheeger isoperimetric inequality is closely related to the project of Ancona on the space of positive harmonic functions of Gromov-hyperbolic manifolds and graphs ( |
and ). In fact, in the study of the Laplace operator on a hyperbolic manifold or graph [MATH] Ancona obtained in these three last papers interesting results, under the additional assumption that the bottom of the spectrum of the Laplace spectrum [MATH] is positive. The well-known Cheeger inequality [MATH] , where [MATH... |
for a converse inequality). Hence, the results of this paper are useful in order to obtain these Ancona’s results. Given any Riemannian [MATH] -manifold [MATH] , the Cheeger isoperimetric constant of [MATH] is defined as |
[EQUATION] where [MATH] ranges over all non-empty bounded open subsets of [MATH] and [MATH] denotes the [MATH] -dimensional Riemannian volume in [MATH] of the set [MATH] We write [MATH] and [MATH] |
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