Convex programs with an additional constraint on the product of several convex functions
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.
scientific article; zbMATH DE number 639284
scientific article