A Class Of Generalized Semi-infinite Programming Problem Transformation And Algorithm

Semi-infinite programming has wide applications in many fields such as engineering design,optimal control,information technology and economic equilibrium.It has become an active field of research in optimization.Recently,with the development of high technology and the profound research on the social economy,a large number of mathematical models of generalized semi-infinite programming emerge in above fields.Therefore,it is very significant to study generalized semi-infinite programming.Because many important results in theoretical and numerical aspects of common semi-infinite programming problems are obtained,we can transform the generalized semi-infinite problems into common semi-infinite problems or finite problems.In particular,using augmented Lagrange functions or penalty functions is the main method to complete the equivalent transformation.In this paper,we study a form of general semi-infinite programming and reformulate the KKT system of the GSIP problems into a system of smooth equations. Given the main method of the solving equations—L-M method which is one type of Newton method,we solve the equations by it.This paper is composed of three chapters.Chapter 1 is the introduction of this paper,which introduces the development of semi-infinite programming and the main results in this paper.In Chapter 2,we extend the type of reformulation of SIP problems to the GSIP problems[30-32].First we reformulate GSIP problem into a system under some conditions at the set X~* that is the set of minimizes of GSIP problems.Then by introducing one form of NCP functions we reformulate the KKT system into a system of semi-smooth equations and we change the form of FB function to make sure continuously differentiable of the KKT system and the sequence generate by L-M method in reference[29]which is the method to solve the nonlinear equations converge to the solution set which is the merit function's staionary point of the smooth function quadratically under a local error bound of the system of smoothing equations.At last we illustrate the good performance of the method by numerical results.But because the Jacobian matrix nonsingularity condition is weaken to be the local error bound condition.If we can guarantee the nonsingularity condition in the level setΩthe solution we get is the stationary point of the GSIP problems.In chapter 3,we solve the KKT system of the GSIP problems by another method. By using a perturbed Fisher-Burmeister function we reformulate the system of semi-smooth equations into a system of smooth equations and we design a smoothing L-M method for solving this system.Then we prove the method is globally and under a local error bound condition for the system of smooth equations the method is superlinearly convergent.Compare with the reference[30,31]our system is a invariable-dimensional system but the system of the paper[30,31]is a variable-dimensional system because p varies with x and we use the assumption of local error bound instead of the nonsingularity condition the level setΩthat is weaker than the nonsingularity condition.At last we illustrate the performance of the method by numerical results.Because of the invariable-dimensional system,the advantage is that we needn't try p=1,then p=2,and so on.Compare with the method in chapter 2 it proves a better performance by numerical method.And the times it need is less although the smoothing is just superlinear.