The filled function transformations for constrained global optimization
type: Item
instance of: scholarly article
title: The filled function transformations for constrained global optimization
zbMATH Open document ID: 0714.65065
DOI: 10.1016/0096-3003(90)90118-M
author: Renpu Ge
published in: Applied Mathematics and Computation
publication date: 1990-01-01T00:00:00Z
review text: The purpose of this paper is to introduce the main idea of the filled function transformation. One shows that there exist several ways to make global optimization (1) global-min f(x) subject to \(c_ i(x)=0\), \(i=1,...,m'\); \(c_ i(x)\geq 0\), \(i=m'+1,...,m.\) The first is to use the exact penalty function \[ P_ e(x,\mu)=f(x)+\sum^{m'}_{i=1}\mu_ i| c_ i(x)| +\sum^{m}_{i=m'+1}\mu_ i\max [0,-c_ i(x)], \] to transform the problem (1) to an unconstrained global minimization problem. The second one constructs filled functions for problem (1) with inequality constraints only. For this, one divides \(R^ n\) into two parts: \(S_ 1=\{x/f(x)\geq f(x^*_ 1)\) or \(x\not\in \Omega_ f\}\), \(S_ 2=\{x/f(x)<f(x^*_ 1)\) and \(x\in \Omega_ f\}\), \((\Omega_ f\) is the feasible region of problem (1)) and one finds a filled function transformation \[ \bar U(x,A,h)=\eta (\| x-x_ 0\|)\phi [A(\max \{[f(x)-f(x^*_ 1)], \quad \max [-c_ i(x)]\}+h)], \] which has no stationary point in \(S_ 1\), but does have a minimizer in \(S_ 2.\) The third way is to use a square penalty function \[ F(x,\mu)=f(x)+\sum^{m'}_{i=1}\mu_ i[c_ i(x)]^ 2+\sum^{m}_{i=m'+1}\mu_ i[\max (0,c_ i(x))]^ 2 \] to transform the problem (1) to a sequence of unconstrained global optimization problems. The fourth way is to solve a sequence of new constrained local minimization problems with filled functions and a part of the original constraints.
reviewed by: Eugenia Duca
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 4176370
zbMATH Keywords: filled function transformation
zbMATH Keywords: global optimization
zbMATH Keywords: penalty function
zbMATH Keywords: unconstrained global minimization problem
MaRDI profile type: Publication
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full work available at URL: 0096-3003(90)90118-m
OpenAlex ID: W2029446757
description: scientific article; zbMATH DE number 4176370