Semi-infinite programming and semi-definite optimization problems

Abstract

The purpose of this thesis is to study combined semi-infinite and semi-definite programming problems (SISDP), generalized semi-infinite programming problenls (GSIP) and optimization problems with max-min constraints. For (SISDP), we derive uniform dualities and zero duality gap properties between the problem (SISDP) and its Lagrangian-type dual problem. We first derive necessary and sufficient conditions for uniform dualities of both the homogeneous (SISDP) problem and the nonhomogeneous (SISDP) problem. Under a generalized canonical closedness condition, we establish uniform duality properties for (SISDP) problem. Moreover, we show that a zero duality gap exists between the problem(SISDP) and its dual problem if Slater's constraint qualification holds. We prove the closedness property of the feasible set mapping for the parametric problem of (SISDP). We obtain a sufficient condition for the upper and lower semi-continuity of the value function of the parametric problem of (SISDP), We also investigate the lower semicontinuity of the value function of the dual parametric problem. On the Other hand, by assuming the continuity of the value function, we investigate the closedness, uniform compactness and upper semicontinuity of the solution set mapping for the parametric problem of (SISDP). We also show that the solution set mapping for its dual parametric problem is uniformly compact. Next, we develop two discretization algorithms, each with an adaptive scheme, for solving(SISDP) problem. We obtain the convergence results of both the algorithms. Finally, we apply these discretization algorithms to solve semi-infinite quadratically constrained quadratic programming, semi-infinite eigenvalue, the continuous-time envelope-constrained filtering and robust envelope-constraind filtering problems. The numerical results obtained illustrate the effectiveness and efficiency of the proposed methods. For (GSIP), an auxiliary optimization problem is introduced. The relationship between local (respectively, global) optimal solutions of the problem (GSIP) and local (respectively global) optimal solution of the auxiliary optimization problem is obtained. By using the idea of the pattern search method, an algorithm is derived for solving the problem (GSIP). Under the convexity conditions of the objective function and the constraint set, we prove that the sequence generated by our algorithm is convergent with its limiting point satisfying a Fritz-John optimality condition. Numerical results obtained show that the algorithm is efficient. An optimization problem with maximin constraints is considered. It is known that the optimization problem is equivalent to a standard nonlinear optimization problem in the sense that a local minimizer of one problem will give rise to a local minimizer of the other problem. We show that equivalent relationship between the two optimization problems is valid under weaker condition. Then, we develop a descent algorithm for solving the optimization problem with maximin constraints. Under the convexity conditions of the objective function and the constraint functions, we prove that the sequence generated by our algorithm is finite and the solution obtained when our algorithm stops is a local optimal solution. Numerical results are given to illustrate the effectiveness of the proposed algorithm.