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(a) Initial population represented within solution space; (b) Initial population represented within objective space
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In [5] an evolutionary algorithm for detecting continuous Pareto optimal sets has been proposed. In this paper we propose a new evolutionary elitist approach combing a non-standard solution representation and an evolutionary optimization technique. The proposed method permits detection of continuous decision regions. In our approach an individual (...
Citations
... The approximation of the entire Pareto front is presented as concatenation of local approximations. For single variable MOOPs, Dumitrescu et al. (2001) proposed an evolutionary method, continuous Pareto set (CPS) algorithm. It uses only a unique mutation operator. ...
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.
In the past few years, multi-objective optimization algorithms have been extensively applied in several fields including engineering design problems. A major reason is the advancement of evolutionary multi-objective optimization (EMO) algorithms that are able to find a set of non-dominated points spread on the respective Pareto-optimal front in a single simulation. Besides just finding a set of Pareto-optimal solutions, one is often interested in capturing knowledge about the variation of variable values over the Pareto-optimal front. Recent innovization approaches for knowledge discovery from Pareto-optimal solutions remain as a major activity in this direction. In this article, a different data-fitting approach for continuous parameterization of the Pareto-optimal front is presented. Cubic B-spline basis functions are used for fitting the data returned by an EMO procedure in a continuous variable space. No prior knowledge about the order in the data is assumed. An automatic procedure for detecting gaps in the Pareto-optimal front is also implemented. The algorithm takes points returned by the EMO as input and returns the control points of the B-spline manifold representing the Pareto-optimal set. Results for several standard and engineering, bi-objective and tri-objective optimization problems demonstrate the usefulness of the proposed procedure.
