Search

Reference sets are widely used by many multi-objective evolutionary algorithms (MOEAs) to decompose the objective space, define search directions, or calculate quality indicators (QIs) embedded into the selection mechanisms. Well-known MOEAs adopt the generation of uniformly distributed points on a unit simplex to construct such reference sets. Although these mechanisms are useful for approximating Pareto fronts with regular shapes, i.e., simplex-like shapes, they have difficulties representing Pareto fronts with irregular geometries. To overcome this drawback, many reference set adaptation methods have been proposed so far. However, some adaptation methods present a degraded performance on regular Pareto front shapes, while others promote a balanced performance. Nevertheless, an extensive assessment has not been made. In this paper, a MOEA based on the inverted generational distance plus indicator, using an adaptive reference set, is used to study the performance of well-known adaptation methods. Although an adaptation method promotes a balanced performance on both regular and irregular Pareto front shapes, results show some difficulties related to the distribution of solutions in complex Pareto front shapes. The results of this study allow detecting the main drawbacks of adaptation methods, which can be addressed by using diversity-oriented selection mechanisms in the generation of reference sets. Hence, these could impact the generation of reference set-based MOEAs achieving good coverage, convergence, and diversity regardless of the Pareto front shape.

In recent years, several indicator-based multi-objective evolutionary algorithms (IB-MOEAs) have been proposed. Each IB-MOEA presents different search preferences depending on the quality indicator (QI) that it uses in its selection mechanism. However, due to these search biases, IB-MOEAs behave differently on each multi-objective optimization problem, producing Pareto front approximations whose characteristics are related to the QI on which they are based. In this paper, we propose a novel algorithm based on the island model that aims to take advantage of the cooperation of individual IB-MOEAs based on the indicators hypervolume, R2, IGD+, +, and Δ p with the aim of improving both convergence and distribution of the Pareto fronts produced. Our experimental results, taking into account seven quality indicators, empirically show that the cooperation of several IB-MOEAs is better than using panmictic versions of them. Additionally, we also show that the performance of our proposal does not depend on the Pareto front shape of the problem being solved.

Commonly, Ant Colony Optimization algorithms have been applied to the solution of single-and multi-objective optimization problems (MOPs). However, in recent years, a number of approaches have been proposed to solve problems with continuous search spaces. One remarkable proposal is the indicator-based Multi-Objective Ant Colony Optimizer for continuous search spaces (iMOACO R ) which is based on the ACO R algorithm and the $R2$ performance indicator, aiming to solve continuous many-objective optimization problems (i.e., MOPs having more than three objective functions). In previous work, we presented an extension of iMOACOR, called iMOACOR-R, in which a recombination operator is employed for solution construction with a fixed, externally-specified probability. In the present work, we introduce a further adaptive variation, called iMOACOR-AR, in which the frequency of applying recombination is dynamically adapted based on the recent past performance of the recombination operator. Our proposal is compared to iMOACOR and iMOACOR-R using 64 standard problems from the multi-objective optimization literature with a number of objectives ranging from 3 to 10. Experimental results show that iMOACOR-AR outperforms iMOACOR and iMOACOR-R in most of the test problems.