Comparando Algoritmos Evolutivos Baseados em Decomposição para Problemas de Otimização Multiobjetivo e com Muitos Objetivos

Many real-world problems can be mathematically modeled as Multiobjective Optimization Problems (MOPs), as they involve multiple conflicting objective functions that must be minimized simultaneously. MOPs with more than 3 objective functions are called Many-objective Optimization Problems (MaOPs). MOPs are typically solved through Multiobjective Evolutionary Algorithms (MOEAs), which can obtain a set of non-dominated optimal solutions, known as a Pareto front, in a single run. The MOEA Based on Decomposition (MOEA/D) is one of the most efficient, dividing a MOP into several single-objective subproblems and optimizing them simultaneously. This study evaluated the performance of MOEA/D and four variants representing the state of the art in the literature (MOEA/DD, MOEA/D-DE, MOEA/D-DU, and MOEA/D-AWA) in MOPs and MaOPs. Computational experiments were conducted using benchmark MOPs and MaOPs from the DTLZ suite considering 3 and 5 objective functions. Additionally, a statistical analysis, including the Wilcoxon test, was performed to evaluate the results obtained in the IGD+ performance indicator. The Hypervolume performance indicator was also considered in the combined Pareto front, formed by all solutions obtained by each MOEA. The experiments revealed that MOEA/DD performed best in IGD+, and MOEA/D-AWA achieved the highest Hypervolume in the combined Pareto front, while MOEA/D-DE registered the worst result in this set of problems.Muitos problemas oriundos do mundo real podem ser modelados matematicamente como Problemas de Otimização Multiobjetivo (POMs), já que possuem diversas funções objetivo conflitantes entre si que devem ser minimizadas simultaneamente. POMs com mais de 3 funções objetivo recebem o nome de Problemas de Otimização com Muitos Objetivos (MaOPs, do inglês Many-objective Optimization Problems). Os POMs geralmente são resolvidos através de Algoritmos Evolutivos Multiobjetivos (MOEAs, do inglês Multiobjective Evolutionary Algorithms), os quais conseguem obter um conjunto de soluções ótimas não dominadas entre si, conhecidos como frente de Pareto, em uma única execução. O MOEA baseado em decomposição (MOEA/D) é um dos mais eficientes, o qual divide um POM em vários subproblemas monobjetivos otimizando-os ao mesmo tempo. Neste estudo foi realizada uma avaliação dos desempenhos do MOEA/D e quatro de suas variantes que representam o estado da arte da literatura (MOEA/DD, MOEA/D-DE, MOEA/D-DU e MOEA/D-AWA) em POMs e MaOPs. Foram conduzidos experimentos computacionais utilizando POMs e MaOPs benchmark do suite DTLZ considerando 3 e 5 funções objetivo. Além disso, foi realizada uma análise estatística que incluiu o teste de Wilcoxon para avaliar os resultados obtidos no indicador de desempenho IGD+. Também foi considerado o indicador de desempenho Hypervolume na frente de Pareto combinada, que é formada por todas as soluções obtidas por cada MOEA. Os experimentos revelaram que o MOEA/DD apresentou a melhor performance no IGD+ e o MOEA/D-AWA obteve o maior Hypervolume na frente de Pareto combinada, enquanto o MOEA/D-DE registrou o pior resultado nesse conjunto de problemas

Polymer single screw extruder optimization using tchebycheff scalarization method and simulated annealing algorithm

The single screw extrusion optimal design involves the optimization of six criteria that can be efficiently handled by a weighted Tchebycheff scalarization method. The performance of the method has been analyzed for three different methods to generate weight vectors. The experimental results show that the tested strategies provide similar and reasonable solutions and supply a valuable procedure to identify good trade-offs between conflicting objectives.European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734205- H2020-MSCA-RISE-2017. The work has also been supported by FCT – Fundação para a Ciência e Tecnologia within the R&D Units Project Scope: UIDB/00319/2020, UIDB/00013/2020 and UIDP/00013/2020 of CMAT-U

The Kalai-Smorodinski solution for many-objective Bayesian optimization

An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends to grow larger with the number of objectives. We thus propose to focus on a specific solution originating from game theory, the Kalai-Smorodinsky solution, which possesses attractive properties. In particular, it ensures equal marginal gains over all objectives. We further make it insensitive to a monotonic transformation of the objectives by considering the objectives in the copula space. A novel tailored algorithm is proposed to search for the solution, in the form of a Bayesian optimization algorithm: sequential sampling decisions are made based on acquisition functions that derive from an instrumental Gaussian process prior. Our approach is tested on four problems with respectively four, six, eight, and nine objectives. The method is available in the Rpackage GPGame available on CRAN at https://cran.r-project.org/package=GPGame

Multi-objective genetic programming with partial sampling and its extension to many-objective

This paper describes a technique on an optimization of tree-structure data by of multi-objective evolutionary algorithm, or multi-objective genetic programming. GP induces bloat of the tree structure as one of the major problem. The cause of bloat is that the tree structure obtained by the crossover operator grows bigger and bigger but its evaluation does not improve. To avoid the risk of bloat, a partial sampling operator is proposed as a mating operator. The size of the tree and a structural distance are introduced into the measure of the tree-structure data as the objective functions in addition to the index of the goodness of tree structure. GP is defined as a three-objective optimization problem. SD is also applied for the ranking of parent individuals instead to the crowding distance of the conventional NSGA-II. When the index of the goodness of tree-structure data is two or more, the number of objective functions in the above problem becomes four or more. We also propose an effective many-objective EA applicable to such the many-objective GP. We focus on NSGA-II based on Pareto partial dominance (NSGA-II-PPD). NSGA-II-PPD requires beforehand a combination list of the number of objective functions to be used for Pareto partial dominance (PPD). The contents of the combination list greatly influence the optimization result. We propose to schedule a parameter r meaning the subset size of objective functions for PPD and to eliminate individuals created by the mating having the same contents as the individual of the archive set

Quality Indicators for Preference-based Evolutionary Multi-objective Optimization Using a Reference Point: A Review and Analysis

Some quality indicators have been proposed for benchmarking preference-based evolutionary multi-objective optimization algorithms using a reference point. Although a systematic review and analysis of the quality indicators are helpful for both benchmarking and practical decision-making, neither has been conducted. In this context, first, this paper reviews existing regions of interest and quality indicators for preference-based evolutionary multi-objective optimization using the reference point. We point out that each quality indicator was designed for a different region of interest. Then, this paper investigates the properties of the quality indicators. We demonstrate that an achievement scalarizing function value is not always consistent with the distance from a solution to the reference point in the objective space. We observe that the regions of interest can be significantly different depending on the position of the reference point and the shape of the Pareto front. We identify undesirable properties of some quality indicators. We also show that the ranking of preference-based evolutionary multi-objective optimization algorithms depends on the choice of quality indicators