Primal or dual strong-duality in nonconvex optimization and a class of quasiconvex problems having zero duality gap

Abstract

Primal or dual strong-duality (or min-sup, inf-max duality) in nonconvex optimization is revisited in view of recent literature on the subject, establishing, in particular, new characterizations for the second case. This gives rise to a new class of quasiconvex problems having zero duality gap or closedness of images of vector mappings associated to those problems. Such conditions are described for the classes of linear fractional functions and that of quadratic ones. In addition, some applications to nonconvex quadratic optimization problems under a single inequality or equality constraint, are presented, providing new results for the fulfillment of zero duality gap or dual strong-duality.

On weak conjugacy, augmented Lagrangians and duality in nonconvex optimization

“…Nevertheless, for an optimization problem, without convexity conditions, the conventional Lagrangian function may not always guarantee the zero duality gap which may occur between the given (primal) problem and the dual one. On the other hand, the additional assumptions may need to have zero duality gap when the linear Lagrangian is considered (see Ernst and Volle 2013;Goberna et al 2014) for the convex and (Ernst and Volle 2016;Flores-Bazan et al 2017;Jeyakumar et al 2009;Polik and Terlaky 2007;Polyak 1998) for some nonconvex cases).…”