Approximating nondominated sets in continuous multiobjective optimization problems

Abstract

Many important problems in Operations Research and Statistics require the computation of nondominated (or Pareto or efficient) sets. This task may be currently undertaken efficiently for discrete sets of alternatives or for continuous sets under special and fairly tight structural conditions. Under more general continuous settings, parametric characterisations of the nondominated set, for example through convex combinations of the objective functions or -constrained problems, or discretizations-based approaches, pose several problems. In this paper, the lack of a general approach to approximate the nondominated set in continuous multiobjective problems is addressed. Our simulation-based procedure only requires to sample from the set of alternatives and check whether an alternative dominates another. Stopping rules, efficient sampling schemes, and procedures to check for dominance are proposed. A continuous approximation to the nondominated set is obtained by fitting a surface through the points of a discrete approximation, using a local (robust) regression method. Other actions like clustering and projecting points onto the frontier are required in nonconvex feasible regions and nonconnected Pareto sets. In a sense, our method may be seen as an evolutionary algorithm with a variable population size.

PAINT: Pareto front interpolation for nonlinear multiobjective optimization

“…However, when looking at PAINT as a plain approximation method, it can be compared to some existing methods in the literature. This kind of approximation methods can be found in a survey [30] and in papers [2,7,9,16,21]. These methods, however, do not concentrate on the question of how to choose a Pareto optimal solution on the produced approximation as we do.…”

Integration of Directed Searches in Particle Swarm Optimization for Multi-Objective Optimization

“…However, the focus of this paper is to enhance the convergence performance of available vector EAs, rather than to compare the performances of the two aforementioned types of multi-objective optimizers. Third, while a wealth of endeavors is devoted in the direction of finding the true Pareto solutions to address the balance between the conflicts in terms of convergence toward the PF, and the requirement to maintain good diversity in the searched Pareto optimal solutions, only lukewarm efforts are given to the development of approximating techniques of non-dominated sets in continuous multi-objective optimization studies [7]. However, a proper approximation of an efficient or non-dominated set could provide a wealth of useful information to guide the search toward the finding of more and better Pareto optimal solutions with enhanced convergence performances.…”

Multiobjective Estimation of Distribution Algorithm Based on Joint Modeling of Objectives and Variables

“…A successful strategy adopted by many MOEAs (including multi-objective EDAs) [29], [30] is to modify the solution selection and replacement mechanisms and use solution reproduction (model learning and sampling for EDAs) as in single objective optimization. However, as we discuss later, the inclusion of more objectives into the problem can also affect how the new solutions are being generated.…”