Author:  Chen, Zhangyou 
Title:  Optimality conditions of semiinfinite programming and generalized semiinfinite programming 
Degree:  Ph.D. 
Year:  2013 
Subject:  Programming (Mathematics) Mathematical optimization. Hong Kong Polytechnic University  Dissertations 
Department:  Dept. of Applied Mathematics 
Pages:  vi, 111 p. ; 30 cm. 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b2653083 
URI:  http://theses.lib.polyu.edu.hk/handle/200/7282 
Abstract:  Semiinfinite programming has been long an important model of optimization problems, arising from areas such as approximation, control, probability. Generalized semiinfinite programming has also been an active research area with relatively short history. Nonetheless it has been known that the study of a generalized semiinfinite programming problem is much more difficult than a semiinfinite programming problem. The purpose of this thesis is to develop necessary optimality conditions for semiinfinite and generalized semiinfinite programming problems with penalty functions techniques as well as other approaches. We introduce two types of pth order penalty functions (0 < p ≤ 1), for semiinfinite 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 semiinfinite programming problems. This process is also applied to generalized semiinfinite programming problems after being equivalently transformed into standard semiinfinite programming problems. Via the transformation of penalty functions of the lower level problems, we study some properties of the feasible set of the generalized semiinfinite programming problem which is known to possess unusual properties such as nonclosedness, reentrant corners, disjunctive structures, and further establish a sequence of approximate optimization problems and approximate properties for generalized semiinfinite programming problems. We also investigate nonsmooth generalized semiinfinite programming problems via generalization differentiation and derive corresponding optimality conditions via variational analysis tools. Finally, we characterize the strong duality theory of generalized semiinfinite programming problems with convex lower level problems via generalized augmented Lagrangians. 
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