A Nonconvex Duality with Zero Gap and Applications

A duality with zero gap for nonconvex optimization problems is presented. The first class of nonconvex problems, where local optima may not be global, is a quasi-convex minimization over a convex set. For this class a generalized Kuhn–Tucker condition is obtained, and the duality is similar to the Fenchel–Moreau–Rockafellar duality scheme. By the duality, one can reduce the problem to solving a system of convex and quasi-convex inequalities. Unlike the previous developments, these conjugation functionals and dual problems are defined on the dual space and involve no extra parameter. For more general nonconvex problems, such as a quasi-convex maximization over a compact set or a general minimization over the complement of a convex set, a duality with zero gap can be obtained as well. A zero gap in primal-dual pairs allows the development of primal-dual algorithms for nonconvex problems. The primal-dual algorithms are very suitable when the dual problem is simpler than the primal one.