Parametric Approximation of the Pareto Set in Multi‐Objective Optimization Problems

Abstract

In this paper, a methodology for the systematic parametric representation for approximating the Pareto set of multi‐objective optimization problems has been proposed. It leads to a parametrization of the solutions of a multi‐objective optimization problem in the design as well as in the objective space, which facilitates the task of a decision maker in a significant manner. This methodology exploits the properties of Fourier series basis functions to approximate the general form of (piecewise) continuous Pareto sets. The structure of the problem helps in attacking the problem in two parts, linear and nonlinear least square error minimization. The methodology is tested on the bi‐objective and tri‐objective problems, for which the Pareto set is a curve or surface, respectively. For assessing the quality of such continuous parametric approximations, a new measure has also been suggested in this work. The proposed measure is based on the residuals of Karush–Kuhn–Tucker conditions and quantifies the quality of the approximation as a whole, as it is defined as integrals over the domain of the parameter(s). Copyright © 2014 John Wiley & Sons, Ltd.

Ensuring Smoothly Navigable Approximation Sets by Bézier Curve Parameterizations in Evolutionary Bi-objective Optimization

“…Optimizing only the control points of the Bézier curve, that define its curvature, enforces the decision variables of solutions in the approximation set to vary in a smooth, continuous fashion, thereby likely improving intuitive navigability of the approximation set. Previous work on parameterizations of the approximation set has been applied mainly in a post-processing step after optimization, or was performed in the objective space [3,15,19], but this does not aid in the navigability of the approximation set in decision space. Moreover, fitting a smooth curve through an already optimized set of solutions might result in a bad fit, resulting in a lower-quality approximation set.…”

Ensuring smoothly navigable approximation sets by Bezier curve parameterizations in evolutionary bi-objective optimization -- applied to brachytherapy treatment planning for prostate cancer

“…Optimizing only the control points of the Bézier curve, that define its curvature, enforces the decision variables of solutions in the approximation set to vary in a smooth, continuous fashion, thereby likely improving intuitive navigability of the approximation set. Previous work on parameterizations of the approximation set has been applied mainly in a post-processing step after optimization, or was performed in the objective space [17,3,24], but this does not aid in the navigability of the approximation set in decision space. Moreover, fitting a smooth curve through an already optimized set of solutions might result in a bad fit, resulting in a lower-quality approximation set.…”