Optimality conditions of semi-infinite programming and generalized semi-infinite programming

Abstract

Semi-infinite programming has been long an important model of optimization problems, arising from areas such as approximation, control, probability. Generalized semi-infinite programming has also been an active research area with relatively short history. Nonetheless it has been known that the study of a generalized semi-infinite programming problem is much more difficult than a semi-infinite programming problem. The purpose of this thesis is to develop necessary optimality conditions for semi-infinite and generalized semi-infinite programming problems with penalty functions techniques as well as other approaches. We introduce two types of p-th order penalty functions (0 < p ≤ 1), for semi-infinite programming problems, and explore various relations between them and their relations with corresponding calmness conditions. Under the exactness of certain type penalty functions and some other appropriate conditions especially second order conditions of the constraint functions, we develop optimality conditions for semi-infinite programming problems. This process is also applied to generalized semi-infinite programming problems after being equivalently transformed into standard semi-infinite programming problems. Via the transformation of penalty functions of the lower level problems, we study some properties of the feasible set of the generalized semi-infinite programming problem which is known to possess unusual properties such as non-closedness, re-entrant corners, disjunctive structures, and further establish a sequence of approximate optimization problems and approximate properties for generalized semi-infinite programming problems. We also investigate nonsmooth generalized semi-infinite programming problems via generalization differentiation and derive corresponding optimality conditions via variational analysis tools. Finally, we characterize the strong duality theory of generalized semi-infinite programming problems with convex lower level problems via generalized augmented Lagrangians.