Exact penalty in d. c. programming (Q1573084)

In this paper a nonconvex program (P) is considered. The objective function \(f\) is concave and the feasible set is the intersection of a polyedral convex set \(K\) with a lower level set of a concave function \(g\). A list of nonconvex optimization problems that can be formulated as (P) is presented. A useful tool to make more tractable (P) is to transform the problem into an equivalent one in the d.c. optimization framework, by penalizing the reverse convex constraint. In the case where \(g\) is nonnegative, the exact penalty approach for (P) is nothing but the Lagrangean duality relative to this problem; within this setting, the authors discuss relationships between problem (P), the penalized problems and the d.c. problem associated to the Lagrangean duality relative to (P). Their main result proves that, if \(g\) is nonnegative and \(K\) is bounded, the last two problems are equivalent. In the end, they prove that solving (P) is equivalent to solving the problem over the smaller feasible set of the vertices of \(K\).