Convergence and diversity of solutions play an essential role in the design of multi-objective evolutionary algorithms (MOEAs). Among the available diversity mechanisms, the e-dominance has shown a proper balance between convergence and diversity. When using e-dominance, diversity is ensured by partitioning the objective space into boxes of size e and, typically, a single solution is allowed at each of these boxes. However, there is no easy way to determine the precise value of ϵ. In this paper, we investigate how this goal can be achieved by using a co-evolutionary scheme that looks for the proper values of e along the search without any need of a previous user's knowledge. We include the proposed co-evolutionary scheme into an MOEA based on ϵ-dominance giving rise to a new MOEA. We evaluate the proposed MOEA solving standard benchmark test problems. According to our results, it is a promising alternative for solving multi-objective optimization problems because three main reasons: 1) it is competitive concerning stateof-the-art MOEAs, 2) it does not need extra information about the problem, and 3) it is computationally efficient.

One of the major limitations of evolutionary algorithms based on the Lebesgue measure for multi-objective optimization is the computational cost required to approximate the Pareto front of a problem. Nonetheless, the Pareto compliance property of the Lebesgue measure makes it one of the most investigated indicators in the design of indicator-based evolutionary algorithms (IBEAs). The main deficiency of IBEAs that use the Lebesgue measure is their computational cost which increases with the number of objectives of the problem. On this matter, the investigation presented in this paper introduces an evolutionary algorithm based on the Lebesgue measure to deal with box-constrained continuous multi-objective optimization problems. The proposed algorithm implicitly uses the regularity property of continuous multi-objective optimization problems that has suggested effectiveness when solving continuous problems with rough Pareto sets. On the other hand, the survival selection mechanism considers the local property of the Lebesgue measure, thus reducing the computational time in our algorithmic approach. The emerging indicator-based evolutionary algorithm is examined and compared versus three state-of-the-art multi-objective evolutionary algorithms based on the Lebesgue measure. In addition, we validate its performance on a set of artificial test problems with various characteristics, including multimodality, separability, and various Pareto front forms, incorporating concavity, convexity, and discontinuity. For a more exhaustive study, the proposed algorithm is evaluated in three real-world applications having four, five, and seven objective functions whose properties are unknown. We show the high competitiveness of our proposed approach, which, in many cases, improved the state-of-the-art indicator-based evolutionary algorithms on the multi-objective problems adopted in our investigation.

One of the main disadvantages of Evolutionary Multi-Objective Algorithms (EMOAs) based on hypervolume is the computational cost of the hypervolume computation. This deficiency gets worse either when an EMOA calculates the hypervolume several times or when it is dealing with problems having more than three objectives. In this sense, some researchers have designed strategies to reduce the number of hypervolume calculations. Among them, the use of the locality property of the hypervolume has emerged as an alternative to deal with this problem. This property states that if a solution is moving in its neighborhood, only its contribution is affected and the contributions of the rest of the solutions remain the same. In this paper, we present a novel evolutionary approach that exploits the locality property of the hypervolume. The proposed approach adopts a probability to use two or three individuals in its environmental selection procedure. In this way, it only needs to compute two or three hypervolume contributions per iteration. The proposed algorithm is evaluated by solving standard benchmark test problems and two real-world applications where the features of the problems are unknown. According to results, the proposed approach is a promising alternative for solving problems with a high number of objectives because three main reasons: i) it is competitive with respect to state-of-the-art EMOAs based on hypervolume, ii) it does not need extra information about the problem (which is particularly essential when solving real-world applications), and iii) its computational cost is much lower than the other hypervolume-based EMOAs.