Study On Optimality Conditions For Semi-infinite Programming And Optimization Methods With Perturbations

Semi-infinite programming deals with optimization problems in which either the number of decision variables or the number of constraints is infinite. It has wide applications in many fields such as economic equilibrium, optimal control, information technology and computer network. With the development of high technology and the profound change of the social economy, a large number of mathematical models of generalized semi-infinite programming emerge in these fields, in which the optimal value functions are generalized by non-compact sets or set-value mappings instead of constant compact sets. Therefore, it is very significant to study the generalized semi-infinite programming.Line-search methods are a class of important numerical methods for nonlinear optimization problems. Many researchers in optimization area dedicate themselves to constructing effective line-search methods.This dissertation is devoted to the study of the first-order optimality conditions for generalized semi-infinite max-min programming with a non-compact set and the perturbation methods for unconstrained optimization problems. The main results, obtained in this dissertation, may be summarized as follows:1. In chapter 2, the differential property of the optimal value function with a non-compact set is studied. The expression of its lower-Hadamard directional derivative is obtained. And the expression of its subdifferential is developed in the case that the effective domain of the optimal value function is a non-empty convex set. Furthermore, the first-order optimality condition and its equivalent reformulations for generalized semi-infinite max-min programming with a non-compact set are presented using the lower-Hadamard directional derivative and subdifferential.2. Chapter 3 studies the gradient-type methods for unconstrained optimization problems. Section 1 proposes a new class of three-term memory gradient methods. The global convergence property of the method is established. Furthermore, in order to improvethe convergence property of the method, a new class of memory gradient projection methods is presented with the property that the whole sequence of iterates converges to a solution to the problem under the conditions such as pseudo-convexity and continuous differentiability of objective function. In section 2, two new classes of methods, called gradient-type method with perturbations and hybrid projection method with perturbations, are proposed. In these methods, non-monotone line search technique is employed, which makes them easily executed in computer. Under mild conditions, the global convergence of these methods is proved as well. Numerical examples are given to show the effectiveness of these methods.3. In chapter 4 and 5, the conjugate gradient methods with perturbations are studied. In chapter 4, three Fletcher-Reeves (abbr. FR) conjugate gradient methods with perturbations are proposed. The global convergence property of the first method is proved under the condition of main directions' sufficient descent. Whereas, in the proof of the convergence for the other two methods, we only need main directions' descent. Importantly and quite interesting, boundedness conditions such as objective function being bounded below, boundedness of level set are not needed. Chapter 5 presents a version of Dai-Yuan (abbr. DY) conjugate gradient method with perturbations. In this method, Wolfe or Armijo stepsize rule is used. The global convergence of the method is proved under mild conditions. Numerical examples indicate that this method is effective.