An Exponent Lagrangian For Solving Semi-in Finite Programming Problem

Many practical problems in the fields of engineering design, optimal control, information technology and economic equilibrium etc, can be modeled as a semi-infinite programming (SIP) problem, which has become a powerful tool to solve practical problems in resent years. Methods for solving SIP problems have been paid more attention. Turning semi-infinite programming problem into nonlinear optimization problem with finite constraints is one of the representative methods. Nonlinear Lagrange methods play an important role in solving constrained optimization problems. This dissertation aims at transforming SIP problem into nonlinear programming with finite constraints, discussing an Exponent Lagrange function for solving the SIP problem and corresponding first and second order optimality conditions. Those of generalized semi-infinite programming (GSIP) problem are discussed as well. Moreover, the relationships between SIP problem and GSIP problem are analyzed. The main results obtained in this dissertation may be summarized as follows:Chapter 2 discusses Exponent Lagrange function method for solving semi-infinite programming problem. Firstly, the conditions for transforming SIP problem into nonlinear programming with finite constraints are analyzed. Nonlinear Lagrange multiplier and Exponent 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 base on exponent Lagrangian are discussed. Finally, an example is presented to illustrate necessary condition for the existence of nonlinear Lagrange multiplier.Chapter 3 discusses an Exponent Lagrangian for solving GSIP problem. It presents the standard Lagrangian of the lower level problem and defines an Exponent Lagrange function of the upper level problem. First-order and second-order optimality conditions for GSIP problem are investigated based on proposed Lagrangian.Section 4 discusses the conditions of transforming the GSIP problem into the SIP problem. It proves that the transformation can be done (at least theoretically) under appropriate assumptions, such as compactness and linear independent constraint qualification(LICQ) is satisfied by set Y (x). On the other hand, it has been shown that the transformation is feasible if function Y ((x|-)) satisfies the M-F constraint qualification and the feasible set Y (x)of lower problem is homeomorphic to function Y ((x|-)) when x converges to (x|-) .