A generalized nonconvex duality with zero gap and applications

Abstract

In a quasiconvex minimization over a convex set although a local minimum may not global, we obtain a convex type duality scheme. In convex type problems, the optimality criterion at a feasible solution z is of the form $$0 \in A(z)$$ (11) where A(z) is a convex set depending on z in the dual space. In nonconvex type problems the global optimality criterion at a feasible solution z is of the form $$C(z) \subset {\rm A}(z)$$ (12) where C(z), A(z) are convex sets depending on z in the dual space (see Thach 10). Of course criterion (12) becomes criterion (11) when C(z) is reduced to {0}. Thus, in some senses, criterion (12) is a generalization of criterion (11). But the duality obtained from (12) is quite different from the duality obtained from (11). The duality obtained from (12) is of the form max=−min (max=max or min=min), whereas the duality obtained from (11) is of the form min=max (max=−max or min=−min). By the duality we can reduce a convex type optimization problem to solving a system of inequations but we could not do the similar thing for a nonconvex type optimization problems.