The Research On Algorithms For Mathematical Programming With Equilibrium Constraints

Mathematical programming with equilibrium constraints contains equality constraints, inequality constraints and complementarity constraints. It is an arousing general interest problem that recently developed in mathematical programming field. These kinds of problems are widely used in economy, engineering design, game theory and so on. More and more people pay attention to this problem in recent years. The achievements can be summarized into the following two aspects.For a class of optimization problems with linear complementarity constraints, we firstly proposed an sequential systems of linear equations (SSLE) algorithm. By using a smoothing complementarity function with arbitrary smoothing parameter, a new smoothing technology is produced. At each iteration of the proposed method, by solving only three systems of linear equations with the same coeffcients, we can get the search direction. In order to avoid Maratos effect, we use the Watchdog technique. When iteration is large enough, we can reduce the calculated steps of algoritnm. Under some suitable assumptions, algorithm is global convergence and superlinear convergence.Secondly, for (MPEC) problem with equality constraints, inequality constraints and complementarity constraints, we combine a gradient projection matrix with penalty function technique to give a project variable metric algorithm with arbitrary initial point. The direction is combined with the descent direction, the feasible direction and the revised direction. Part of the direction is combined with the feasible direction and the revised direction. Furthermore, global convergence and superlinear convergence of the proposed method is proved.In the end, we make some numerical experiments to the above algorithms, and the results show that they are effective.