Luis V. Santana-Quintero

Pareto-adaptive ε-dominance

Efficiency has become one of the main concerns in evolutionary multiobjective optimization during recent years. One of the possible alternatives to achieve a faster convergence is to use a relaxed form of Pareto dominance that allows us to regulate the granularity of the approximation of the Pareto front that we wish to achieve. One such relaxed forms of Pareto dominance that has become popular in the last few years is -dominance, which has been mainly used as an archiving strategy in some multiobjective evolutionary algorithms. Despite its advantages, -dominance has some limitations. In this paper, we propose a mechanism that can be seen as a variant of -dominance, which we call Pareto-adaptive -dominance (pa-dominance). Our proposed approach tries to overcome the main limitation of -dominance: the loss of several nondominated solutions from the hypergrid adopted in the archive because of the way in which solutions are selected within each box.

A Review of Techniques for Handling Expensive Functions in Evolutionary Multi-Objective Optimization

Evolutionary algorithms have been very popular for solving multiobjective optimization problems, mainly because of their ease of use, and their wide applicability. However, multi-objective evolutionary algorithms (MOEAs) tend to consume an important number of objective function evaluations, in order to achieve a reasonably good approximation of the Pareto front. This is a major concern when attempting to use MOEAs for real-world applications, since we can normally afford only a fairly limited number of fitness function evaluations in such cases. Despite these concerns, relatively few efforts have been reported in the literature to reduce the computational cost of MOEAs. It has been until relatively recently, that researchers have developed techniques to achieve an effective reduction of fitness function evaluations by exploiting knowledge acquired during the search. In this chapter, we analyze different proposals currently available in the specialized literature to deal with expensive functions in evolutionary multi-objective optimization. Additionally, we review some real-world applications of these methods, which can be seen as case studies in which such techniques led to a substantial reduction in the computational cost of the MOEA adopted. Finally, we also indicate some of the potential paths for future research in this area.

A new proposal for multi-objective optimization using differential evolution and rough sets theory

This paper presents a new multi-objective evolutionary algorithm (MOEA) based on differential evolution and rough sets theory. The proposed approach adopts an external archive in order to retain the nondominated solutions found during the evolutionary process. Additionally, the approach also incorporates the concept of paε-dominance to get a good distribution of the solutions retained. The main idea of the approach is to use differential evolution (DE) as our main search engine, trying to translate its good convergence properties exhibited in single-objective optimization to the multi-objective case. Rough sets theory is adopted in a second stage of the search in order to improve the spread of the nondominated solutions that have been found so far. Our hybrid approach is validated using standard test functions and metrics commonly adopted in the specialized literature. Our results are compared with respect to the NSGA-II, which is a MOEA representative of the state-of-the-art in the area.

Seeding the initial population of a multi-objective evolutionary algorithm using gradient-based information

In the field of single-objective optimization, hybrid variants of gradient-based methods and evolutionary algorithms have been shown to perform better than an evolutionary method by itself. This same idea has been recently used in Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most cases, gradient information is used along the whole process, which involves a high computational cost, mainly related to the computation of the step lengths required. In contrast, in this paper we propose the use of gradient information only at the beginning of the search process. We will show that this sort of scheme maintains results of good quality while considerably decreasing the computational cost. In our work, we adopt a steepest descent method to generate some nondominated points which are then used to seed the initial population of a multi-objective evolutionary algorithm (MOEA), which will spread them along the Pareto front. The MOEA adopted in our case is the NSGA-II, which is representative of the state-of-the-art in the area. To validate our proposal, we adopt box-constrained continuous problems (the ZDT test suite). The gradients required are approximated using quadratic regressions. Our proposed approach performs a total of 2000 objective function evaluations, which is much lower than the number of evaluations normally adopted with the ZDT test suite in the specialized literature. Our results are compared with respect to the ldquopurerdquo NSGA-II (i.e., without using gradient-based information) so that the potential benefit of these initial solutions fed into the population can be properly assessed.

EMOPSO: a multi-objective particle swarm optimizer with emphasis on efficiency

This paper presents the Efficient Multi-Objective Particle Swarm Optimizer (EMOPSO), which is an improved version of a multi-objective evolutionary algorithm (MOEA) previously proposed by the authors. Throughout the paper, we provide several details of the design process that led us to EMOPSO. The main issues discussed are: the mechanism to maintain a set of well-distributed nondominated solutions, the turbulence operator that avoids premature convergence, the constraint-handling scheme, and the study of parameters that led us to propose a self-adaptation mechanism. The final algorithm is able to produce reasonably good approximations of the Pareto front of problems with up to 30 decision variables, while performing only 2,000 fitness function evaluations. As far as we know, this is the lowest number of evaluations reported so far for any multi-objective particle swarm optimizer. Our results are compared with respect to the NSGA-II in 12 test functions taken from the specialized literature.

A new proposal for multiobjective optimization using particle swarm optimization and rough sets theory

This paper presents a new multi-objective evolutionary algorithm which consists of a hybrid between a particle swarm optimization approach and some concepts from rough sets theory. The main idea of the approach is to combine the high convergence rate of the particle swarm optimization algorithm with a local search approach based on rough sets that is able to spread the nondominated solutions found, so that a good distribution along the Pareto front is achieved. Our proposed approach is able to converge in several test functions of 10 to 30 decision variables with only 4,000 fitness function evaluations. This is a very low number of evaluations if compared with todays standards in the specialized literature. Our proposed approach was validated using nine standard test functions commonly adopted in the specialized literature. Our results were compared with respect to a multi-objective evolutionary algorithm that is representative of the state-of-the-art in the area: the NSGA-II.

A Memetic PSO Algorithm for Scalar Optimization Problems

In this paper we introduce line search strategies originating from continuous optimization for the realization of the guidance mechanism in particle swarm optimization for scalar optimization problems. Since these techniques are well-suited for-but not restricted to-local search the resulting algorithm can be considered to be memetic. Further, we will use the same techniques for the construction of a new variant of a hill climber. We will discuss possible realizations and will finally present some numerical results indicating the strength of the two algorithms

Knowledge Incorporation in Multi-objective Evolutionary Algorithms

This chapter presents a survey of techniques used to incorporate knowledge into evolutionary algorithms, with a particular emphasis on multi-objective optimization. We focus on two main groups of techniques: those that incorporate knowledge into the fitness evaluation, and those that incorporate knowledge in the initialization process and the operators of an evolutionary algorithm. Several techniques representative of each of these groups are briefly discussed, together with some examples found in the specialized literature. In the last part of the chapter, we provide some research ideas that are worth exploring in the future by researchers interested in this topic.

Use of Radial Basis Functions and Rough Sets for Evolutionary Multi-Objective Optimization

This paper presents a new multi-objective evolutionary algorithm (MOEA) which adopts a radial basis function (RBF) approach in order to reduce the number of fitness function evaluations performed to reach the Pareto front. The specific method adopted is derived from a comparative study conducted among several RBFs. In all cases, the NSGA-II (which is an approach representative of the state-of-the-art in the area) is adopted as our search engine with which the RBFs are hybridized. The resulting algorithm can produce very reasonable approximations of the true Pareto front with a very low number of evaluations, but is not able to spread solutions in an appropriate manner. This led us to introduce a second stage to the algorithm in which it is hybridized with rough sets theory in order to improve the spread of solutions. Rough sets, in this case, act as a local search approach which is able to generate solutions in the neighborhood of the few nondominated solutions previously generated. We show that our proposed hybrid approach only requires 2,000 fitness function evaluations in order to solve test problems with up to 30 decision variables. This is a very low value when compared with todays standards reported in the specialized literature

A multi-objective particle swarm optimizer hybridized with scatter search

This paper presents a new multi-objective evolutionary algorithm which consists of a hybrid between a particle swarm optimization (PSO) approach and scatter search. The main idea of the approach is to combine the high convergence rate of the particle swarm optimization algorithm with a local search approach based on scatter search. We propose a new leader selection scheme for PSO, which aims to accelerate convergence. Upon applying PSO, scatter search acts as a local search scheme, improving the spread of the nondominated solutions found so far. Thus, the hybrid constitutes an efficient multi-objective evolutionary algorithm, which can produce reasonably good approximations of the Pareto fronts of multi-objective problems of high dimensionality, while only performing 4,000 fitness function evaluations. Our proposed approach is validated using ten standard test functions commonly adopted in the specialized literature. Our results are compared with respect to a multi-objective evolutionary algorithm that is representative of the state-of-the-art in the area: the NSGA-II.