Managing Radial Basis Functions for Evolutionary Many-Objective optimization

This paper proposes a radial basis functions (RBFs) assisted evolutionary algorithm for solving expensive many-objective problems where only a small number of real fitness evaluations are permitted. Two kinds of RBFs are applied in this algorithm, and the differences between the two kinds of RBFs are figured out to provide the estimated errors. By doing this, the estimated individual which has the maximum difference will be evaluated by real functions to strengthen the RBF models. In addition, for each objective, a more suitable RBF is selected for the purpose of making a more accurate approximation of the real functions. The simulation results demonstrate that the proposed algorithm not only performs well on many-objective problems with 10 decision variables, but also shows high efficiency. Besides, the proposed algorithm has good performance on problems with up to 30 decision variables.

I. Introduction

Multi-objective optimization problems (MOPs) appear widely in real world. For MOPs, what we can obtain is a set of trade-off solutions called Pareto Front (PF) rather than a single optimal solution, since the conflicting nature of each objectives. An MOP can be formulated as $$\begin{equation*}\begin{array}{l} \min f(\mathbf{x})=\left(f_{1}(\mathbf{x}), f_{2}(\mathbf{x}), \ldots, f_{M}(\mathbf{x})\right) \\ \text {s.t. } \mathbf{x} \in \mathbf{X} \end{array} \tag{1}\end{equation*}$$