A Logarithmic Lagrangian For Solving Semi-infinite Programming Problem

Semi-infinite programming(SIP) is a research topic with important theoretical significance and practical value in the field of mathematic programming. Many significant problems in engineering, economics, management, information technology, computer network systems and so on, such as robot path problem, production design planning, air pollution control, etc, are semi-infinite programming problems. Methods for solving SIP problems have been paid more attention in areas of optimization. Turning the semi-infinite programming problem into nonlinear optimization problem with finite constraints is one of the representative methods. The dual method based on nonlinear Lagrange function requires no restrictions on the feasibility of primal variables. Nonlinear Lagrange methods play an important role in solving constrained optimization problems. This dissertation is devoted to the study of Logarithmic Lagrangians for solving semi-infinite programming and generalized semi-infinite programming. The main results obtained in this dissertation may be summarized as follows:Chapter 2 discusses Logarithmic Lagrange function method for solving semi-infinite programming problem. Firstly, nonlinear Lagrange multiplier and logarithmic Lagrange function of semi-infinite programming problem are defined, and the duality theory based on proposed Lagrangian is analyzed. Secondly, first and second order optimality conditions based on logarithmic Lagrangian are proved. Finally, the necessary conditions for the existence of nonlinear Lagrange multipliers are illustrated by an example.Chapter 3 discusses Logarithmic Lagrange function method for solving generalized semi-infinite programming problem. Firstly, the logarithmic Lagrange function of generalized semi-infinite programming problem is defined. Secondly, first and second order optimality conditions based on proposed Lagrangian are proved.Chapter 4 analyzes the relationship between semi-infinite programming problem and generalized semi-infinite programming problem. It is feasible that transforming the generalized semi-infinite programming into the semi-infinite programming problem under M-F constraint qualification. Furthermore, the conditions for transformation are explored and proved.