Theory And Algorithm For Mathematical Programs With Complementarity Constraints

In this paper,we are mainly concerned with the numerical algorithm for the mathematical programs with complementarity constraints.The existence of complementary conditions prevents the classical theories and algorithms of nonlinear programming problems from being directly applied to solve complementary constrained optimization problems,the main energy of the people is concentrated on the dealing with complementary constraint conditions.For example,Huang.etc got the augmented Lagrange function problem without constraints by adding all the constraints to the objective function,and proposed an augmented Lagrange function method,but the objective function form of the problem is complex.Tin.etc proposed the penalty function algorithm for solving complementarity constraints optimization problem by punishing equality complementary constraints,but the convergence of the algorithm was not given.Scholtes obtained sequence slack problems by transforming equality complementary constraints into inequality constraints and proposed the slack method,but did not analyze the solution of the slack problem.Yan.etc used the smoothing functions to deal with the complementarity conditions and proposed a smoothing method.In this paper we do further research on the basis of read literature and then propose a semismooth partial augmented Lagrange function method and a sequence penalty function method based on slack technique for solving complementarity constrained optimization problems.In Chapter 1,we mainly introduce the research history,related theories and methods for complementarity constrained optimization problem and its research significance.We introduce the penalty function method,smoothing method,slack method,augmented Lagrange function method and related convergence results for solving complementarity constrained optimization problems.In Chapter 2,we first use the FB function to transform the complementary conditions into equality constraint conditions,and obtain the equivalent nonlinear programming problem of complementary constrained optimization problems.Then,we analyze the partial augmented Lagrange function problem of the latter,and propose a semismooth partial augmented Lagrange function method.And we study the first-order and second-order optimality conditions of the partial augmented Lagrange function problem,and we give the construction of the algorithm.Finally,we prove that the limit point of the stationary point sequence generated by the algorithm is the B-stationary point of complementarity constrained optimization problem.Compared with the existing augmented Lagrange function method,our method has simpler objective function form or less constraint conditions.In Chapter 3,we first use slack technique to deal with complementarity conditions and obtain the slack problem.Then,we analyze the partial penalty function problem of the slack problem and put forward a sequence penalty function method based on slack technique.And we study the relationship between the optimal solution of the partial penalty function problem and the optimal solution of the slack problem,and give the basic structure of the algorithm.Finally,we prove that the algorithm converges the stationary point sequence to the B-stationary point of the complementarity constrained optimization problem.The sequence penalty function method proposed by us is based on the slack constraints of complementary constraints,and contains the advantages of the slack method with good convergence.In Chapter 4,we test the algorithm,and the results of numerical experiments show that our algorithm is effective.