Convex programs with several additional reverse convex constraints (Q1087136)

This paper addresses the general problem of minimizing a convex function subject to convex and reverse convex constraints. Here by a reverse convex constraint is meant any constraint of the form g(x)\(\leq 0\), where \(g: R^ n\to R\) is a concave function (i.e. g is a convex function). Clearly, a reverse convex constraint determines a closed subset of \(R^ n\), whose complement is an open convex set (therefore, in the literature such constraints are also termed ''complementary convex''. In the present paper we shall focus our attention on the most typical problem of the mentioned class, namely the following problem \[ (P)\quad Minimize\quad f(x),\quad s.t.\quad x\in D,\quad g(x)\leq 0, \] where \(f: R^ n\to R\) is a convex finite function, D is a closed convex subset of \(R^ n\) given by a convex finite function \(h: R^ n\to R\); \(D=\{x\in R^ n:\) h(x)\(\leq 0\}\), where \(g: R^ n\to R\) is a concave finite function.