Scalable and customizable benchmark problems for many-objective optimization

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.This work has been supported by the Brazilian agencies (i) National Council for Scientific and Technological Development (CNPq); (ii) Coordination for the Improvement of Higher Education (CAPES) and (iii) Foundation for Research of the State of Minas Gerais (FAPEMIG, in Portuguese)

Learning to decompose: a paradigm for decomposition-based multiobjective optimization

This is the author accepted manuscript. The final version is available from IEEE via the DOI in this recordThe decomposition-based evolutionary multiobjective optimization (EMO) algorithm has become an increasingly popular choice for a posteriori multiobjective optimization. However, recent studies have shown that their performance strongly depends on the Pareto front (PF) shapes. This can be attributed to the decomposition method, of which the reference points and subproblem formulation settings are not well adaptable to various problem characteristics. In this paper, we develop a learning-to-decompose (LTD) paradigm that adaptively sets the decomposition method by learning the characteristics of the estimated PF. Specifically, it consists of two interdependent parts, i.e., a learning module and an optimization module. Given the current nondominated solutions from the optimization module, the learning module periodically learns an analytical model of the estimated PF. Thereafter, useful information is extracted from the learned model to set the decomposition method for the optimization module: 1) reference points compliant with the PF shape and 2) subproblem formulations whose contours and search directions are appropriate for the current status. Accordingly, the optimization module, which can be any decomposition-based EMO algorithm in principle, decomposes the multiobjective optimization problem into a number of subproblems and optimizes them simultaneously. To validate our proposed LTD paradigm, we integrate it with two decomposition-based EMO algorithms, and compare them with four state-of-the-art algorithms on a series of benchmark problems with various PF shapes.Royal Societ

A Scalable Test Suite for Continuous Dynamic Multiobjective Optimization

Dynamic multiobjective optimization (DMO) has gained increasing attention in recent years. Test problems are of great importance in order to facilitate the development of advanced algorithms that can handle dynamic environments well. However, many of the existing dynamic multiobjective test problems have not been rigorously constructed and analyzed, which may induce some unexpected bias when they are used for algorithmic analysis. In this paper, some of these biases are identified after a review of widely used test problems. These include poor scalability of objectives and, more importantly, problematic overemphasis of static properties rather than dynamics making it difficult to draw accurate conclusion about the strengths and weaknesses of the algorithms studied. A diverse set of dynamics and features are then highlighted that a good test suite should have. We further develop a scalable continuous test suite, which includes a number of dynamics or features that have been rarely considered in literature but frequently occur in real life. It is demonstrated with empirical studies that the proposed test suite are more challenging to the DMO algorithms found in the literature. The test suite can also test algorithms in ways that existing test suites cannot

Peeking beyond peaks: challenges and research potentials of continuous multimodal multi-objective optimization

Computer Systems, Imagery and Medi

On the utilization of pair-potential energy functions in multi-objective optimization

In evolutionary multi-objective optimization (EMO), the pair-potential energy functions (PPFs) have been used to construct diversity-preserving mechanisms to improve Pareto front approximations. Despite PPFs have shown promising results when dealing with different Pareto front geometries, there are still some open research questions to improve the way we employ them. In this paper, we answer three important questions: (1) what is the effect of a crucial parameter of some PPFs?, (2) how do we set the optimal parameter value?, and (3) what is the best PPF in EMO? To solve these questions, we designed a brand-new fast algorithm to generate an approximate solution to a PPF-based subset selection problem and, then, we conducted a comprehensive parametrical study to predict the optimal parameter values using a deep neural network. To show the effectiveness of the PPF-based diversity-preserving mechanisms, we selected two application cases: the generation of reference point sets of benchmark problems (DTLZ, WFG, IDTLZ, IWFG, IMOP, and Viennet) with different Pareto front shapes, and the definition of a PPF-based archive that can be coupled to any multi-objective evolutionary algorithm to construct well-diversified Pareto front approximations. Using several diversity indicators, it is shown that the utilization of PPF-based mechanisms lead to good Pareto front approximations regardless of the Pareto front shape

AREA: An adaptive reference-set based evolutionary algorithm for multiobjective optimisation

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Population-based evolutionary algorithms have great potential to handle multiobjective optimisation problems. However, the performance of these algorithms depends largely on problem characteristics. There is a need to improve these algorithms for wide applicability. References, often specified by the decision maker’s preference in different forms, are very effective to boost the performance of algorithms. This paper proposes a novel framework for effective use of references to strengthen algorithms. This framework considers references as search targets which can be adjusted based on the information collected during the search. The proposed framework is combined with new strategies, such as reference adaptation and adaptive local mating, to solve different types of problems. The proposed algorithm is compared with state-of-the-arts on a wide range of problems with diverse characteristics. The comparison and extensive sensitivity analysis demonstrate that the proposed algorithm is competitive and robust across different types of problems studied in this paper