Scalable and customizable benchmark problems for many-objective optimization

Abstract

Solving many-objective problems (MaOPs) is still a significant challenge in the multi-objective optimization (MOO) field. One way to measure algorithm performance is through the use of benchmark functions (also called test functions or test suites), which are artificial problems with a well-defined mathematical formulation, known solutions and a variety of features and difficulties. In this paper we propose a parameterized generator of scalable and customizable benchmark problems for MaOPs. It is able to generate problems that reproduce features present in other benchmarks and also problems with some new features. We propose here the concept of generative benchmarking, in which one can generate an infinite number of MOO problems, by varying parameters that control specific features that the problem should have: scalability in the number of variables and objectives, bias, deceptiveness, multimodality, robust and non-robust solutions, shape of the Pareto front, and constraints. The proposed Generalized Position-Distance (GPD) tunable benchmark generator uses the position-distance paradigm, a basic approach to building test functions, used in other benchmarks such as Deb, Thiele, Laumanns and Zitzler (DTLZ), Walking Fish Group (WFG) and others. It includes scalable problems in any number of variables and objectives and it presents Pareto fronts with different characteristics. The resulting functions are easy to understand and visualize, easy to implement, fast to compute and their Pareto optimal solutions are known.

Learning Pairwise Comparisons with Machine Learning for Large-Scale Multi-Criteria Decision Making Problems

“…In this paper, it is considered that a typical MCDM approach arranges a finite number of alternatives in a preferable sort way. These preferences can be represented by using a twodimensional matrix as in (1). Also, it is considered a post hoc decision making problems.…”

“…The alternatives used in this work come from the GPD [1]. A many-objective problem was formulated to generate a more complex problem.…”

“…In a deterministic decision problem, the evaluation of the alternative a i under the criterion C j is given by C j (a i ). This decision matrix, or (1), is built at the beginning of the decision process and it represents the nominal values of each alternative in each criterion achieved in the objective space during the optimization.…”

“…These solutions are incomparable, or non-dominated solutions, and often delivered by stochastic heuristic algorithms, particularly the Multi-Objective Evolutionary Algorithms (MOEAs). These solutions can come from benchmark functions (also called test functions or test suites), such as the Generalized Position-Distance (GPD) [1], an intelligent tool for generating scalable and customizable problems in Many-Objective Optimization (MaOP).…”

“…To do so, a small number of solutions is presented per iteration and, once the MAUF is learned, the preferences of the remaining solutions are predicted with the trained model. As a test case, a set of solutions from a many-objective problem formulated with the GPD [1] is used. The expected result can be either one where the ranking does not vary as new solutions are inserted or one where the ranking generated is similar to that of the AHP.…”