Optimality Analysis Of A Class Of Semi-infinite Programming Problems

The semi-infinite programming(SIP)problem is very common in real applications such as engineering design and game theory.In some application problems,the semi-infinite programming has some special forms,then it is necessary to study the properties and algorithms of some special semi-infinite programming problems.We consider a class of semi-infinite programming problems,where there is a parameter.By analyzing the optimal value of the problem,this paper proposes a novel algorithm to find the limit of the optimal value and optimal solution of the SIP problem.Numerical examples in application problems are used to verify the effectiveness of the algorithm.In the first part,we study the semi-infinite programming problem of single infinite constraint with parameters.First,as the parameter increases,we prove that the optimal values decreases monotonically.And the limit of optimal values exists as the parameter tends to infinity.Then,a new algorithm is proposed to obtain the limit,that is,by treating the vector function of the original problem as the decision variables,we apply decomposition method to set up a series of simplified subproblems,and find the the maximum of the optimal values after solving all the subproblems.Next,we apply the fixed point theorem to prove that the maximum value is exactly the limit of optimal values as the parameter tends to infinity under some conditions.Thus,the limit of optimal values as the parameter tends to infinity can be obtained efficiently by solving a series of simplified subproblem.Finally,the theory is verified by numerical examples.In the second part,we study the semi-infinite programming problem of multiple infinite constraints with parameters.First,the optimal value series is proved to be monotonic and the limit exists,and then we apply the decomposition method to set up a series of subproblems.Next,by the condition of LICQ constraint qualification,we prove that the maximum value of the optimal values of the subproblems is equal to the limit value.Finally,by using numerical examples,the optimal objective function value is obtained by increasing parameters and the proposed method respectively,and the optimal value and running time are compared show the efficiency of the proposed method.In the third part,the application of semi-infinite programming in minimax optimization problem is studied.We consider the filter design problem of microphone array.We set up the minimax optimization problem with parameters,and transform it into a semi-infinite programming problem equivalently.We apply the decomposition method to solve this problem.The results show that the proposed algorithm can solve a class of nonsmooth minimax optimization problems efficiently and effectively.Finally,we summarize the work above and make a prospect for future research.