A Convex Optimization Toolbox
This chapter presents the duality theory for optimization problems, by both the minimax and perturbation approach, in a Banach space setting. Under some stability (qualification) hypotheses, it is shown that the dual problem has a nonempty and bounded set of solutions. This leads to the subdifferential calculus, which appears to be nothing but a partial subdifferential rule. Applications are provided to the infimal convolution, as well as recession and perspective functions. The relaxation of some nonconvex problems is analyzed thanks to the Shapley–Folkman theorem.