An optimal bound for d.c. programs with convex constraints

Abstract

A well-known strategy for obtaining a lower bound on the minimum of a d.c. function f−g over a compact convex set S⊂ℝn consists of replacing the convex function f by a linear minorant at x 0∈S. In this note we show that the x 0 * giving the optimal bound can be obtained by solving a convex minimization program, which corresponds to a Lagrangian decomposition of the problem. Moreover, if S is a simplex, the optimal Lagrangian multiplier can be obtained by solving a system of n + 1 linear equations.