Convex programs with an additional constraint on the product of several convex functions
type: Item
instance of: scholarly article
title: Convex programs with an additional constraint on the product of several convex functions
zbMATH Open document ID: 0827.90114
DOI: 10.1016/0377-2217(94)90375-1
author: Takahito Kuno
author: Yasutoshi Yajima
author: Yoshitsugu Yamamoto
author: Hiroshi Konno
published in: European Journal of Operational Research
publication date: 1995-12-13T00:00:00Z
review text: The authors discuss optimization problems of the form \[ \text{minimize}\quad f_0(x),\quad\text{subject to}\quad x\in X,\;\prod^p_{i= 1} f_i(x)\leq 1, \] where \(f_i: \mathbb{R}^n\to \mathbb{R}\), \(i= 0, 1, \dots, p\), are convex functions (with \(f_i> 0\) for \(i= 1,\dots, p\)) and \(X\subseteq \mathbb{R}^n\) is a compact convex set. Using the convex marginal function \[ g(\xi):= \inf\{f_0(x)\mid x\in X,\;f_i(x)\leq \xi_i,\;i= 1,\dots, p\} \] (here the values of \(g(\xi)\) are solutions of convex optimization problems) it turns out that the original problem can be reduced to the \(p\)-dimensional reverse convex problem \[ \text{minimize}\quad g(\xi),\quad\text{subject to}\quad \xi\geq 0,\;\prod^p_{i= 1} \xi_i\leq 1. \] The authors propose two outer approximation algorithms for solving the resulting problem, in which the feasible set is approximated by the union of rectangles and the union of simplices respectively. In case of \(p\leq 4\), computational experiments indicate good results for obtaining \(\varepsilon\)-feasible solutions. Some remarks regarding the extension to more general problems in which the product function is replaced by a composite function \(h(f_1(x),\dots, f_p(x))\) with a nondecreasing quasiconvex function \(h: \mathbb{R}^p\to \mathbb{R}\) are given at the end of the paper.
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 90C26
Mathematics Subject Classification ID: 49M37
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 639284
zbMATH Keywords: product of convex functions
zbMATH Keywords: reverse convex problem
zbMATH Keywords: outer approximation algorithm
reviewed by: Jörg Thierfelder
MaRDI profile type: Publication
full work available at URL: 0377-2217(94)90375-1
OpenAlex ID: W1980686870
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description: scientific article; zbMATH DE number 639284