A Kriging Model-Based Expensive Multiobjective Optimization Algorithm Using R2 Indicator of Expectation Improvement

Ding Han,Jianrong Zheng

doi:10.1155/2020/9474580

Abstract

Most of the multiobjective optimization problems in engineering involve the evaluation of expensive objectives and constraint functions, for which an approximate model-based multiobjective optimization algorithm is usually employed, but requires a large amount of function evaluation. Aiming at effectively reducing the computation cost, a novel infilling point criterion EIR2 is proposed, whose basic idea is mapping a point in objective space into a set in expectation improvement space and utilizing the R2 indicator of the set to quantify the fitness of the point being selected as an infilling point. This criterion has an analytic form regardless of the number of objectives and demands lower calculation resources. Combining the Kriging model, optimal Latin hypercube sampling, and particle swarm optimization, an algorithm, EIR2-MOEA, is developed for solving expensive multiobjective optimization problems and applied to three sets of standard test functions of varying difficulty and comparing with two other competitive infill point criteria. Results show that EIR2 has higher resource utilization efficiency, and the resulting nondominated solution set possesses good convergence and diversity. By coupling with the average probability of feasibility, the EIR2 criterion is capable of dealing with expensive constrained multiobjective optimization problems and its efficiency is successfully validated in the optimal design of energy storage flywheel.

Highlights

Science and engineering practice possess a large number of multiobjective optimization problems (MOP) [1] whose objectives often conflict with each other and need to be optimized simultaneously, where the Pareto set is desired
Expectation Improvement R2 Indicator. e main difficulty in extending EI infilling criterion to MOPs lies in the fact that we have to specify ymin before calculating the EI value, which is impractical as no such solution exists as being optimal with respect to every objective, but a set of nondominated solution found during previous optimization process
TEST1, TEST2, and FON are all biobjective test functions but have, respectively, convex, piecewise continuous, and concave Pareto front. e comparison of mean and std value for HV, Inverted Generational Distance (IGD), and Nondominated solution ratio (NR) metric can be found at the top part of Tables 3–5, respectively

Summary

Introduction

Science and engineering practice possess a large number of multiobjective optimization problems (MOP) [1] whose objectives often conflict with each other and need to be optimized simultaneously, where the Pareto set is desired. Objective functions and constraint functions may come from finite element simulation; they have no explicit analytic expressions and are usually computationally expensive to evaluate. Under this condition, we are interested in identifying a set of nondominated solutions, called approximated Pareto Set/Front, to represent the true Preto Set/Front, consuming as less computing resources as possible. Kriging model is an unbiased estimation model with smallest estimated variance, excellent high-dimensional, and nonlinear fitting capability. It assumes the relationship between input x and predicted output y􏽢 as follows: y􏽢(x) μ + z(x),

Methods

Discussion

Conclusion