Optimality conditions and duality relations in nonconvexoptimization

This thesis aims to establish conditions for the objective and the constraint functions such that the zero duality gap condition is satisfied without convexity assumptions. In the literature, the conditions given on the objective and the constraint functions for guaranteeing the zero duality gap condition are still an unsolved problem. We investigate the zero duality gap condition for a class of nonconvex optimization problems. In this study, a positively homogeneous and lower semicontinuous objective and constraint functions are defined on a conic set and the zero duality gap condition is obtained by showing the perturbation function is weakly subdifferentiable at the origin. Also, the weak subdifferentiability theorem is proven for the Lipschitz functions. This theorem is based on the separation theorem in the space where epigraph is defined. To prove this theorem augmented dual cones are extended to a higher dimension since epigraph is defined in R^(n+1). Additionally, we investigate the sum rule for the weak subdifferential. It is shown that the weak subdifferentials of some classes of the Clarke directionally differentiable functions and the tangentially convex functions, satisfy this property in the form of equality. A relationship between the weak subdifferential of the indicator function and the augmented normal cone to a nonconvex set is revealed.