Penalty-function-free Methods For Semi-infinite Programming

Semi-infinite programming(SIP)problem is an optimization problem in which finitely many variables appear in infinitely many constraints.This model naturally arises in an abundant number of applications in different fields,such as mathematical physics,robotics,engineering design,filter design,robust optimization,air pollution control,electricity control,economics,etc.In this paper,the effective numerical methods are proposed for semi-infinite programming.In this paper,there are two main work for general SIP and convex SIP problems.For the general SIP,the nonmonotone filter method is proposed.In this method,the SIP problem can be transformed into a nonlinear programming,then to a system of semismooth equations by nonlinear complementary function.Only one linear equations is needed to slove at each iteration,so that computational scale is reduced.Besides,we modify the multidimensional filter which is used for unconstrained optimization to one dimensional filter.Also,the Lagrangian multiplier is adopted in the filter set.So the penalty parameter is avoided in this algorithm.On the other hand,the filter acceptable criterion for a trial point is relaxed by using the nonmonotone strategy,which reduces the Maratos effect to a certain degree.There is another work for convex SIP problem.A bundle method without penalty function is presented for convex semi-infinite programming problem.After transforming the SIP problem to finite programming,we use bundle information approximating the subgradient of the nonsmooth function to obtain a quadratic programming subproblem.The modified filter technique is used to determine whether a trial point is accepted or not.Moreover,based on the nonmonotone technique,the sufficient descent conditions of objective function and constraint violation function are given.Compared with the previous methods,objective function in this method is allowed to be increased in finite steps.Under the suitable conditions,the global convergence is proved.Then numerical experiments are reported,which shows that the effectiveness of this algorithm.