# Convex programs with an additional reverse convex constraint

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## Abstract

A method is presented for solving a class of global optimization problems of the form (P): minimize*f*(*x*), subject to*x*∈*D*,*g*(*x*)≥0, where*D* is a closed convex subset of*R*^{ n } and*f*,*g* are convex finite functions*R*^{ n }. Under suitable stability hypotheses, it is shown that a feasible point\(\bar x\) is optimal if and only if 0=max{*g*(*x*):*x*∈*D*,*f*(*x*)≤*f*(\(\bar x\))}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblems*Q*_{ k },*k*=1, 2, ..., each of which consists in maximizing the convex function*g*(*x*) over some polyhedron*S*_{ k }. The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.

## Key Words

Reverse convex constraints convex maximization concave minimization outer approximation methods## Preview

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© Plenum Publishing Corporation 1987