A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization

Recently research into many-objective optimization has attracted much attention. One of the main topics of the research is to develop evolutionary many-objective optimization (EMAO) algorithms that can solve optimization problems with more than three objectives. EMAO algorithms generally differ from evolutionary multi-objective optimization (EMO) algorithms as EMO algorithms are known to work mainly for optimization problems with two or three objectives. Thus far the performance of EMO algorithms has been validated using both benchmark test problems and real-world applications. Although the performance of EMAO algorithms has also been shown using benchmark test problems, their performance on real-world applications rarely appears in the literature. in this paper we examine the performance of state-of-the-art EMAO algorithms by applying them to a real-world application, namely a hybrid car controller optimization problem with six objectives. It is demonstrated that EMAO algorithms work well for this optimization problem.

A novel hybrid many-objective evolutionary algorithm called Reference Vector Guided Evolutionary Algorithm based on hypervolume indicator (H-RVEA) is proposed in this paper. The reference vectors are used in a number of sub-problems to decompose the optimization problem. An adaptation strategy is used in the proposed algorithm to adjust the reference vector distribution. The proposed algorithm is compared over well-known benchmark test functions with five state-of-the-art evolutionary algorithms. The results show H-RVEA’s superior performance in terms of the inverted generational distance and hypervolume performance measures than the competitor algorithms. The suggested algorithm’s computational complexity is also analysed. The statistical tests are carried out to demonstrate the statistical significance of the proposed algorithm. In order to demonstrate its efficiency, H-RVEA is also applied to solve two real-life constrained many-objective optimization problems. The experimental results indicate that the proposed algorithm can solve the many-objective real-life problems. Note that the source codes of the proposed technique are available at http://dhimangaurav.com/.

In Pareto-based many-objective evolutionary algorithms, performance usually degrades drastically as the number of objectives increases due to the poor discriminability of Pareto optimality. Although some relaxed Pareto domination relations have been proposed to relieve the loss of selection pressure, it is hard to maintain good population diversity, especially in the late phase of evolution. To solve this problem, we propose a symmetrical Generalized Pareto Dominance and Adjusted Reference Vectors Cooperative (GPDARVC) evolutionary algorithm to deal with many-objective optimization problems. The symmetric version of generalized Pareto dominance (GPD), as an efficient framework, provides sufficient selection pressure without degrading diversity, no matter of the number of objectives. Then, reference vectors (RVs), initially generated evenly in the objective space, guide the selection with good diversity. The cooperation of GPD and RVs in environmental selection in part ensures a good balance of convergence and diversity. Also, to further enhance the effectiveness of RV-guided selection, we regenerate more RVs according to the proportion of valid RVs; thereafter, we select the most valid RVs for adjustment after the association operation. To validate the performance of GPDARVC, we compare it with seven representative algorithms on commonly used sets of problems. This comprehensive analysis results in 26 test problems with different objective numbers and 6 practical problems, which show that GPDARVC outperforms other algorithms in most cases, indicating its great potential to solve many-objective optimization problems.

Decomposition‐based evolutionary multiobjective algorithms (MOEAs) divide a multiobjective problem into several subproblems by using a set of predefined uniformly distributed reference vectors and can achieve good overall performance especially in maintaining population diversity. However, they encounter huge difficulties in addressing problems with irregular Pareto fronts (PFs) since many reference vectors do not work during the searching process. To cope with this problem, this paper aims to improve an existing decomposition‐based algorithm called reference vector‐guided evolutionary algorithm (RVEA) by designing an adaptive reference vector adjustment strategy. By adding the strategy, the predefined reference vectors will be adjusted according to the distribution of promising solutions with good overall performance and the subspaces in which the PF lies may be further divided to contribute more to the searching process. Besides, the selection pressure with respect to convergence performance posed by RVEA is mainly from the length of normalized objective vectors and the metric is poor in evaluating the convergence performance of a solution with the increase of objective size. Motivated by that, an improved angle‐penalized distance (APD) method is developed to better distinguish solutions with sound convergence performance in each subspace. To investigate the performance of the proposed algorithm, extensive experiments are conducted to compare it with 5 state‐of‐the‐art decomposition‐based algorithms on 3‐, 5‐, 8‐, and 10‐objective MaF1–MaF9. The results demonstrate that the proposed algorithm obtains the best overall performance.

Decomposition-based evolutionary multi-objective algorithms (MOEAs) and many-objective algorithms (MaOEAs) divide a multi-objective problem (MOP) or a many-objective problem (MaOP) into several subproblems by using a set of predefined uniformly distributed reference vectors and can achieve good overall performance especially in maintaining population diversity. However, they encounter huge difficulties in addressing problems with irregular Pareto Fronts (PFs) since many reference vectors do not work during the searching process. To cope with this problem, this paper aims to improve an existing decomposition-based algorithm called reference vector guided evolutionary algorithm (RVEA) by designing an adaptive reference vectors adjustment strategy and strengthening the poor selection pressure. By adding the adaptive strategy, the predefined reference vectors will be dynamically adjusted according to the distribution of promising solutions with good overall performance and the subspaces where the PF lies may be further divided so as to contribute more to the searching process. Besides, the selection pressure with respect to convergence performance posed by RVEA is mainly from the length of normalized objective vectors and the metric is poor in evaluating the convergence performance of a solution with the increasing of objective size. Motivated by that, an improved angle-penalized distance (APD) method based on a newly proposed fractional dominance relation is developed to better distinguish solutions with sound convergence performance in each subspace. To investigate the performance of the proposed algorithm, extensive experiments are conducted to compare it with 5 state-of-the-art decomposition-based algorithms on 3-, 5-, 8-, 10- objective MaF1-MaF9. The results demonstrate that the proposed algorithm obtains the best overall performance.

In the evolutionary multi-objective optimization, keeping a good balance between convergence and diversity is particularly important to the performance of evolutionary algorithms. In addition, it is becoming more and more important to bring into user preferences, as the number of targets increases, the possibility of using a limited population to achieve a representative subset of the Pareto optimal solution will be reduced. A multi-objective optimization evolutionary algorithm guided by reference vector is proposed. The reference vector can not only be used to decompose the original multi-objective optimization problem into several single-objective subproblems, but also can be used to clarify user preferences for a preference subset in the whole Pareto frontier.

Evolutionary algorithms have shown their promise in coping with many-objective optimization problems. However, the strategies of balancing convergence and diversity and the effectiveness of handling problems with irregular Pareto fronts (PFs) are still far from perfect. To address these issues, this paper proposes an adaptive sorting-based evolutionary algorithm based on the idea of decomposition. First, we propose an adaptive sorting-based environmental selection strategy. Solutions in each subpopulation (partitioned by reference vectors) are sorted based on their convergence. Those with better convergence are further sorted based on their diversity, then being selected according to their sorting levels. Second, we provide an adaptive promising subpopulation sorting-based environmental selection strategy for problems which may have irregular PFs. This strategy provides additional sorting-based selection effort on promising subpopulations after the general environmental selection process. Third, we extend the algorithm to handle constraints. Finally, we conduct an extensive experimental study on the proposed algorithm by comparing with start-of-the-state algorithms. Results demonstrate the superiority of the proposed algorithm.

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The development of algorithms to solve Many-objective optimization problems (MaOPs) has attracted significant research interest in recent years. Solving various types of Pareto front (PF) is a daunting challenge for evolutionary algorithm. A Research mode based evolutionary algorithm (RMEA) is proposed for many-objective optimization. The archive in the RMEA is used to store non-dominated solutions that can reflect the shape of the PF to guide the reference vector adaptation. Information concerning the population is collected, once the number of non-dominated solutions reaches its limit after many generations without exceeding a given threshold, RMEA introduces a research mode that generates more reference vectors to search through the solutions. The proposed algorithm showed competitive performance with four state-of-the-art evolutionary algorithms in a large number of experiments.

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