A proximal method for composite minimization

Abstract. We consider minimization of functions that are compositions of prox-regular functions with smooth vector functions. A wide variety of important optimization problems can be formulated in this way. We describe a subproblem constructed from a linearized approximation to the objective and a regularization term, investigating the properties of local solutions of this subproblem and showing that they eventually identify a manifold containing the solution of the original problem. We propose an algorithmic framework based on this subproblem and prove a global convergence result

Methods Of Variational Analysis In Pessimistic Bilevel Programming

Bilevel programming problems are of growing interest both from theoretical and practical points of view. These models are used in various applications, such as economic planning, network design, and so on. The purpose of this dissertation is to study the pessimistic (or strong) version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation developed by B. Mordukhovich. This dissertation begins with analyzing pessimistic bilevel programs, formulation of the problems, literature review, practical application, existence of the optimal solutions, reformulation and related to the other programming. The mainstream in studying optimization problems consists of obtaining necessary optimality conditions for optimality, and the main focus of this dissertation is to obtain necessary optimality conditions for pessimistic bilevel programming problems. Optimality conditions for the optimistic version of bilevel programming are extensively discussed in the literature. However, there are just a few papers devoted to the pessimistic version of bilevel programming problems and most of these papers concern the existence of optimal solutions. This dissertation is devoted to establish, by a variety of techniques from convex and nonsmooth analysis, several versions of first order necessary and sufficient optimality conditions for pessimistic bilevel programming problems. To achieve our goal, we first use the implicit programming techniques, and depending on the continuous, Lipschitz, and Fréchet differentiable selections, we obtain necessary optimality conditions. The value function technique plays a central role in sensitivity analysis, controllability, and even in establishing necessary optimality conditions. We consider constructions or estimations of the subdifferential of value functions and come up with the optimality conditions using minimax programming approach treating the cases: convex data, differentiable (strict) data, and Lipschitz data separately. We also use the duality programming approach and obtain optimality conditions extending the convex case to the nonconvex case. In the last chapter, we produce the necessary and sufficient optimality conditions for pessimistic bilevel programming with the rational reaction (optimal solutions set of the lower level problem) set of finite cardinality, but not singleton. We present then some classes of pessimistic bilevel programs for which there are finite rational responses