text stringlengths 128 2.05k |
|---|
Non-/Quantifiable A constraint is said to be quantifiable if its degree of feasibility and/or constraint violation can be determined. Otherwise, the constraint is denoted nonquantifiable . That is, nonquantifiable constraints may only return a boolean expression regarding a constraint’s feasibility. |
Un-/Relaxable Unrelaxable constraints define conditions for the parameter vectors that are required to be satisfied to obtain meaningful outputs from either objective functions or simulations. In contrast, relaxable constraints represent desired conditions which do not have to be satisfied at each stage of the optimiza... |
A priori/Simulation In case that the feasibility of a constraint can be evaluated directly, it is referred to as a priori constraint. A constraint that requires running a simulation to verify its feasibility is denoted a simulation constraint. |
Known/Hidden While hidden constraints are unknown to a solver, known constraints are explicitly stated in the problem formulation and thus available to the solver. Notice, hidden constraints are distinctive of simulation-based optimization problems. They are nonquantifiable and unrelaxable by definition. |
All combinations of the above categories are reasonable for the definition of problem instances for a constrained benchmark problem. However, the benchmark suites considered in this paper are usually dedicated to known a-priori , and quantifiable constraints. Whether some constraints are relaxable or unrelaxable is dep... |
for a more detailed explanation. Note that, by interpreting all constraints as unrelaxable , the problem instances would become considerably harder for EA to satisfy. That is, suitable algorithms would have to be equipped with a sophisticated repair technique that allows generating usable candidate solutions in every s... |
The real-valued constrained optimization problems (COP) considered in this report have the general representation [EQUATION] In this context, [MATH] denotes the [MATH] -dimensional search space parameter vector. The set [MATH] usually comprises a number of box-constraints specifying reasonable intervals of the paramete... |
[EQUATION] where the vector [MATH] consists of the component-wise lower bounds, and [MATH] the vector of upper bounds, respectively. Note that, [MATH] is understood as the component-wise less than or equal inequality. The set [MATH] is also referred to as the box of problem ( COP ). |
The feasible region of the search space is additionally restricted by [MATH] real-valued constraint functions. These constraint functions are separated into [MATH] inequality constraints [MATH] , and [MATH] |
equality constraints [MATH] . A vector [MATH] that satisfies all constraints is called feasible. The set of all feasible parameter vectors is referred to as |
[EQUATION] The global optimum of ( COP ) is denoted by [MATH] Note that the objective function [MATH] subject to some constraints is also referred to as constrained function Multiple representations of one specific constrained function that are subject to small variations are denoted as instances of that respective con... |
The box-constraints which impose restrictions on the parameter vector components are usually considered unrelaxable. On the contrary, inequality and equality constraints are considered relaxable insofar as the constrained functions can be evaluated for infeasible parameter vectors and such infeasible candidate solution... |
The size of the feasible region [MATH] relative to the box size [MATH] is denoted by [EQUATION] The parameter [MATH] can be estimated by uniformly sampling a sufficiently large number of candidate solutions inside the set [MATH] and by counting the feasible candidate solutions among these, as suggested in |
Considering problem ( COP ), evolutionary algorithms employ a measure of infeasibility to guide the search process into feasible regions of the search space. The constraint violation [MATH] of a candidate solution [MATH] is usually specified as |
[EQUATION] Multiple definitions of the constraint violation measure [MATH] can be found in the literature, and the choice of which definition to use is essentially left to the search algorithm. EA commonly use [MATH] to create penalty functions, to derive appropriate repair terms, or to rank infeasible candidate soluti... |
[EQUATION] with functions [MATH] and [MATH] defined by [EQUATION] and [EQUATION] In contrast to classical deterministic solvers, equality constraints cause real difficulties for meta-heuristics like EAs. In order to enable EA to satisfy the equality constraints at least up to a fair degree, Eq. ( ) introduces the error... |
Having obtained a notion of feasibility and infeasibility of candidate solutions allows for the introduction of a corresponding order relation. Such order relations permit the comparison of both feasible and infeasible candidate solutions. A commonly used order relation in the field of constrained optimization is the l... |
[MATH] which is defined in a very intuitive way. Two solutions are compared at a time according to the following criteria: 1. Any feasible solution is preferred to an infeasible solution. |
2. Among two feasible solutions, the one having the better objective function value is considered superior. 3. Two infeasible solutions are ranked according to their constraint violation value (the lower the better). |
In mathematical form, this order relation reads [EQUATION] Introduced in , the concept of the lexicographic ordering [MATH] is also referred to as superiority of feasible solutions . The presented order relation is commonly used for the ranking of algorithm realizations in the CEC benchmarks |
Principles for EA benchmarks on constrained optimization tasks Having introduced the general problem formulation in Sec. , this section is concerned with the collection of requirements and preferable features that have to be taken into account when creating a credible benchmark problem (or even framework) for constrain... |
, experimental rigor , and the presentation style of results obtained are appended. The section is divided into three parts: the fundamental principles of the test environment, the design of adequate experiments, and the reporting of test results. Overlaps of these concepts cannot entirely be avoided. |
In many cases, it is not possible to give a final recommendation of the best practice. Hence, it is not within the scope of this article to provide definitive answers to these questions, but rather to create categories that allow a comparative study of distinct benchmark environments. Ultimately, benchmarking suites ar... |
Taking into account publications that report on benchmarking results, ignorance of some of these principles is frequently observed. Hence, this survey may also serve as a (by no means exhaustive) checklist to support authors and/or reviewers of such papers. |
3.1 Fundamental principles of a test environment Each set of benchmark problems should ensure reproducibility of the results obtained by a specific algorithm as well as the comparability of outcome |
Problem domain and documentation A benchmark suite that covers all conceivable features of constrained optimization problems and their combinations appears unmanageable. Hence, it is recommended that a benchmark design systematically focuses on a specific problem subdomain instead of collecting a vast amount of arbitra... |
Well-developed benchmarking environments are supposed to guide the user through the benchmarking process. Users should receive clear instructions regarding the correct use of the benchmark environment, its working principles, the related benchmarking conventions, and the required reporting style. This calls for the cle... |
Problem publicity It is to some degree necessary to decide whether the analytical description of a single problem instance is openly available or whether it is generated at random. The first case allows the user to obtain a notion of the problem complexity. Further, it facilitates the incorporation of real-world proble... |
Such issues can be partly circumvented by generating individual instances of a fixed constrained optimization problem at random. This involves the implementation of an elaborated test-case generator. Due to the complexity of instantiation of real-world applications, this comes with the need for designing suitable artif... |
, the user should not at all be involved in the evaluation of the constrained function. To this end, the benchmark collection would need to provide an easily and freely accessible software environment that offers well-defined input/output specifications. The availability of interfaces to multiple programming languages ... |
Function evaluations It is imperative to provide a clear policy of how to count objective function evaluations and constraint evaluations, respectively. A first option is to interpret the evaluation of the whole constrained function, i.e. the evaluation of the objective function as well as all related constraints, as o... |
The finest-grained approach would be accounting the objective function and all real-valued constrained functions separately. By proper aggregation, this would still allow to use recent presentation styles (refer to Sec. and Sec. ). It might further reveal insights into algorithm working principles on specific problems ... |
Box-constraints A recommendation for the treatment of box-constraints needs to be stated to ensure reproducibility and comparability of the algorithm results. |
According to , its absence may have significant implications on the comparability of algorithm results. In the respective paper, it was pointed out that different box-constraint handling interpretations can produce dissimilar outcomes even for a single algorithm. The study distinguished three box-constraint scenarios: |
(S1) unrelaxable box-constraints, (S2) relaxable box-constraints, and (S3) no box constraints at all. While scenario (S3) is self-explanatory, the box-constraints are defined and enforced at any stage of the search process in situation (S1). Candidate solutions outside the box are considered invalid and thus have to be... |
that algorithms were sometimes able to find solutions of better quality when facing situation (S2) or (S3) instead of (S1), and even if the global optimizer was not located on the boundary of the specified box [MATH] |
In order to avoid inconsistencies, various options come to mind. First, the box-constrained treatment can be completely eliminated if the admissible intervals of the parameter vector components [MATH] are directly included in the inequality constraints [MATH] . In case of one specific lower and upper bound for each par... |
3.2 Experimental design The experimental design of a benchmark testbed is supposed to properly reflect the characteristics of the chosen problem (sub)domain. This requires the unambiguous description of the constrained test problems, initialization practices, as well as appropriate quality indicators. The benchmark pro... |
Initialization Differences with respect to the initialization parameter vectors are present. These have varying implications on the applicability of certain optimizers. A benchmark problem might either provide a feasible initial candidate solution, supply a subset of not necessarily feasible parameter vectors (e.g. by ... |
Precision Considering randomized algorithms, a test environment needs to make assumptions on the termination precision and reasonable error margins for constraint satisfaction. The latter is particularly important in the context of equality constraints because it is otherwise highly improbable to find feasible candidat... |
Actually, although it is commonly done, the consideration of relative precisions (or absolute precisions in the case of [MATH] ) of order [MATH] or even smaller does not always reflect the needs of real-world optimization problems. That is, at some point of the search process the effort to realize very small improvemen... |
Constrained problems A sufficient number of profound constrained optimization problems suitable to represent the chosen problem domain need to be appointed. The problems might either be automatically generated or collected from test problem collections. Each problem needs to be specified in the manner of ( COP ). That ... |
Taking into account that current algorithms have to deal with continuously increasing problem complexity, the constrained functions are ideally designed in a scalable fashion |
. Scalability with respect to the search space dimension, and also the number of constraint functions, permits an understanding of the inherent problem complexity. It further allows assessing these factors of influence on the algorithm performance. In this regard, the creation of artificial test problems represents a m... |
Building clusters of constrained problems with similar features facilitates insight into the algorithm performance on each of the problem subgroups. It further supports the decision whether an algorithmic idea is useful when dealing with specific real-world applications of a certain characteristic |
. For example, regarding a practical application that involves satisfying a great number of constraints, algorithms that have been observed to perform well on test problem subgroups with similar features are of interest. These are usually expected to be better suited than the collectively best algorithm which ultimatel... |
Moreover, the design of problem instances preferably should exclude biases towards certain algorithm classes. To this end, problem formulations aligned in the Cartesian axes should be avoided. Further, problems whose optimum is located on the boundary of the box [MATH] may exhibit the tendency to favor EA that use spec... |
Order relation Benchmark environments that compare algorithms on the basis of solution quality need a consistent order relation for ranking the provided candidate solution realizations. To this end the order relation should be able to deal with feasible and infeasible candidate solutions. A commonly used approach is th... |
which is recapped in Eq. ( ). Benchmarking environments might take into account different ordering instructions. However, these need to be motivated convincingly. |
Quality indicators Multiple aspects of algorithm performance have to be covered by the experimental design The benchmark environment has to use a number of well-defined quality indicators that are computed in the experiments. The quality indicators reflect the suitability of a respective algorithm for a specific constr... |
To obtain the quantity of variability, benchmarking of randomized algorithms involves running multiple independent algorithmic runs on the same problem instances. The appropriate number of repetitions is connected to the choice of quality indicators |
. In order to obtain reasonable statistics a minimum number of 10 to 25 algorithm runs is usually recommended. Termination A benchmark collection might determine strict rules on the termination conditions for participating algorithms, e.g. a fixed budget of function evaluations. Another approach would be to set multipl... |
3.3 Reporting This section takes into account useful principles that support a reproducible and comprehensible presentation of obtained algorithm results. Further, it is concerned with the aspect of algorithm comparison and mentions the need for encouraging algorithm developers to thoroughly report algorithmic details. |
Newsworthiness and presentation To ensure that meaningful results are generated, the benchmarking environment can support the user by providing a performance baseline. Such a baseline may represent performance results obtained by application of comparable algorithms for constrained optimization. If a collection of algo... |
The performance results have to be presented in informative ways to support the interpretation of the individual algorithmic behavior. This is preferably realized by stipulating a presentation style that uses a combination of tables and figures. By providing aggregated algorithm results for the complete benchmark colle... |
Ranking of algorithms Alongside with the presentation of individual algorithm performance, it is the purpose of a benchmark environment to answer the question which algorithm is best suited for solving (a subset of) the benchmark problems. The comparability of the algorithmic results is ensured by defining an appropria... |
Regarding constrained benchmarking functions, the comparability of algorithms results is in need of an ordering approach that is able to distinguish between feasible and infeasible realizations of the obtained quality indicators. A suitable representation of such an order relation is provided by the lexicographic order... |
to make use of consensus rankings which comprise more than one order relation. This way, a consensus ranking allows computing an appropriate algorithm ranking over the whole benchmark suite, or subsets of constrained problems, respectively. |
In order to decide whether comparably small performance differences can be considered significant, the algorithm comparison usually benefits from factoring in statistical hypothesis testing. Being less restrictive than parametric approaches and requiring smaller sample sizes, non-parametric tests are usually recommende... |
. However, statistical and practical significance are not necessarily equivalent and a well-established graphical representation of the algorithm results may suffice |
Algorithm description When providing benchmark results, it should be mandatory to require a proper characterization of a tested algorithm. Such a description includes the detailed motivation of prior investigations and a comprehensive description of the implemented algorithmic ideas. Further, an exact pseudo-code repre... |
PC configuration When it comes to measuring the computational running time of an algorithm, the users of benchmark collections should be required to report on the complete PC configuration. This includes detailed information about the processor architecture, memory, operating system, and the programming language, confe... |
The use of performance benchmarks to calibrate algorithm speed is also recommended in order to obtain a perception of the system-depending performance. This way algorithm comparability can be maintained over long periods of time |
Runtime and algorithm complexity Algorithm efficiency can be assessed by accounting the number of resources needed to reach a given high-quality solution. |
To this end, the CPU time (or wall-clock time) needed for a predifed number of elementary operations can be determined. The consumed time provides an estimate of the algorithm complexity. In order to ensure comparable results, baseline measurements are necessary. However, measureing algorithm efficiency by means of CPU... |
According to , a machine-independent performance criterion suitable for direct search algorithms is the algorithm runtime in terms of the number of function evaluations executed. That is, the measurement of CPU time can be regarded irrelevant in the context of derivative-free optimization. This approach assumes the ava... |
Further, benchmark suites may concentrate on the computation of different indicators like mean or median solution quality. Such studies may argue that their focus is limited on the effectiveness of the algorithms and that runtime can be neglected in this context. Yet, regardless of the primary goals of a benchmark set,... |
It can be used to indicate algorithm complexity, i.e. running time trade-offs that are related to increased solution quality and vice versa. Further, it provides a notion of the computational effort for reproducing the reported results and may provide useful information for assessing parallelization attempts. |
Anyway, plain instructions for computing the algorithm speed have to be provided. This is achieved by indicating whether the calculations are performed for only one exemplary algorithm run or whether it considers all repetitions. Further, the running time may cover all preprocessing and initialization steps, or it migh... |
The CEC competition on constrained real-parameter optimization The test function sets defined in the context of the IEEE Congress on Evolutionary Computation (CEC) competitions on single objective constrained real-parameter optimization are arguably the most common test collections for benchmarking randomized search al... |
, 2010 , and 2017 . Each of these competitions introduced a specific set of constrained test problems in the line with ( COP ). The test functions sets are supported with a policy for the computation of comprehensive performance indicators and for reporting algorithm results. |
The remainder of this section is concerned with reviewing the benchmarking conventions associated with the mentioned CEC benchmark environments as well as their characteristic features. To this end, the benchmark definitions are examined by taking into account three different aspects: the basic benchmarking conventions... |
4.1 Benchmarking conventions The CEC2006 benchmarks build a test environment of 24 distinct constrained functions with various features. The first 11 constrained problems (p01 to p11) were originally collected in |
, problems p12 and p13 are taken from , problems p21 and p22 can be traced back to heat exchange network applications , p23 was suggested in |
, and p24 can be found in . For the remaining test problems (p14 to p20) it is referred to The succeeding benchmark definitions for CEC2010 |
introduced 18 new constrained benchmark problems. Yet, the origin of the corresponding constrained functions is not easily comprehensible. Only one constrained function was adopted from the CEC2006 benchmarks. The benchmark set introduced variations of 8 distinct objective functions that differ with respect to the appl... |
, e.g. the Rosenbrock function, the Griewank function, and the Weierstrass function. Other function definitions were obtained by use of the test-case generator proposed in |
However, being defined in scalable from with respect to the search space dimension, the constrained test problems have to be solved in dimension [MATH] and [MATH] |
Considering even larger search space dimensions ( [MATH] [MATH] [MATH] , and [MATH] ), a novel collection of 28 benchmark problems was created for the CEC2017 competition |
. The 2017 constrained function definitions are designed by taking new combinations of the building blocks provided in . However, some overlaps do exist. It is claimed that the CEC2006 benchmarks and the CEC2010 have been successfully solved |
. Yet, the older CEC testbeds are still very popular for benchmarking direct search algorithms and particularly Evolutionary Algorithms, e.g. |
In contrast, the CEC2017 problem definitions are reutilized for the CEC competition on single objective constrained real-parameter optimization taking place during the IEEE World Congress on Computational Intelligence (WCCI) in 2018. |
The constrained function definitions are fully presented in the corresponding technical reports. Yet, some constrained problems lack a description of the translation vectors and rotation matrices. These can only be understood by taking into account their implementations. The corresponding code is maintained on the resp... |
. It is openly available in the programming languages C and MATLAB. The consecutive development from CEC2006 towards the CEC2017 benchmarks is not entirely motivated in the corresponding technical reports. Modifications with respect to performance indicators or algorithm ranking approaches are not entirely transparent.... |
All three technical reports of the constrained CEC benchmark collections demand to identify the evaluation of the whole constrained function as one single function evaluation. That is, each constrained function evaluation consumes one function evaluation of the predefined budget regardless of whether the objective func... |
The CEC competitions for constraint real-parameter optimization do not enforce the feasibility of search space parameter vectors. In this respect, equality and inequality constraints of a constrained function ( COP ) are always considered as relaxable, cf. option (S2) in Sec. That is, the algorithms are allowed to move... |
For instance, algorithms might be allowed to solely operate outside the feasible region until the optimizer is approached sufficiently close. |
A specific treatment of box-constraints is not stipulated by the CEC benchmarks. The technical reports are not clear on whether box-constraints have to be regarded relaxable (S2) or unrelaxable (S1). This ambiguity can potentially result in different approaches, and ultimately in significant performance differences |
Taking into account the most successful strategies reported in CEC competitions and after inspecting the related openly available source codes, up to our knowledge, all algorithms were assuming situation (S1) as introduced in Sec. Albeit reporting the full algorithm can be considered scientific standard, yet some paper... |
For the computation of the quality indicators (see Sec. 4.3 ), the CEC framework sorts the algorithm realizations of 25 independent runs on the basis of the lexicographic ordering relation introduced in ( ). That is, feasible solutions are ranked based on their objective function values. They always dominate infeasible... |
4.2 Experimental design The CEC competitions on constrained real-parameter optimization do not provide an initially feasible region or candidate solution. Instead individual box-constraints are specified for each constrained problem and algorithms are supposed to randomly sample a starting point or an initial populatio... |
Hence, the feasibility of initial candidate solutions is not ensured. In order to be competitive on the CEC benchmarks, algorithms need to be able to deal with infeasible solutions. This is affirmed when considering the size of the feasible region [MATH] relative to [MATH] , i.e. the parameter [MATH] (cf. Eq. ( )). Loo... |
Regarding the CEC2006 competition, the detailed benchmark function specifications can be found in the technical report . The benchmark set consists of [MATH] constrained functions of varying search space dimensions between [MATH] and [MATH] The given constrained functions are fixed in terms of the problem dimension and... |
The 2006 benchmarks include [MATH] fully separable constrained functions. Refraining from the use of parameter vector rotations, the benchmarks enclose a potential bias towards strategies that search predominantly along the coordinate axes of the search space |
In this regard, the CEC2006 benchmarks favor algorithms that use coordinate-wise search or differences of obtained candidate solutions, e.g. Coordinate Search or Differential Evolution variants. |
The benchmark definitions of the CEC2010 competition can be considered a refinement with respect to this issue. As mentioned above, the constrained problems of CEC2010 can be affiliated to different sources |
and are partly designed by use of the test-case generator The 2010 competition included [MATH] constrained functions in dimensions [MATH] , and [MATH] , respectively. The formulation of scalable constrained functions allows for conclusions with respect to an algorithm’s ability to deal with growing search space dimensi... |
Still, [MATH] out of [MATH] problems are fully separable and do not apply any rotations to the parameter vectors. While the formal description of those transformations is not satisfactorily explained in the technical report, it is deposited in the corresponding competition source code |
There, the transformations are deterministically specified, and different, for each individual constrained function. Having a look at the CEC2017 competition, the constrained function definitions are quite similar to its predecessor competition. The corresponding technical report |
states [MATH] scalable constrained optimization problems essentially attributable to the same sources of the CEC2010 benchmarks. The latest CEC collection considers not only a larger number of problems but also larger search space dimensions: [MATH] [MATH] [MATH] , and [MATH] In total, the competition comprises [MATH] ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.