Research On Sequential Optimality Conditions And Algorithm For Mathematical Programs With Equilibrium Constraints

Mathematical programs with vertical complementarity constraints(MPVCC)problem is a new form of equilibrium constraint optimization model,it has many applications in the fields of economic equilibrium,engineering and machine learning.The feasible set has a very special structure and violates most of the standard constraint qualifications.Without constraint qualifications,the Karush-Kuhn-Tucker conditions may not hold at minimizer,and thus,the convergence assumptions for several methods for solving constrained optimization problems are not fulfilled.It's necessary to consider suitable optimality conditions,constraints qualifications,and designed algorithms from a theoretical and numerical point of view.In recent years,many researchers pay attention to the sequential optimality condition for nonlinear programming because it is closely related to the stopping criterion of the algorithm and convergence analysis.Therefore,we introduce the sequential optimality condition for MPVCC and present the constraint qualification for M-stationarity associated with sequential optimality condition.We study the sequential optimality condition and convergence analysis of the existing method for MPVCC.The main results as follows:First,we propose some sequential optimality conditions for usual stationarity concepts for MPVCC,namely,AW-stationarity and AM-stationarity.The AM-stationarity is stronger than AW-stationarity.We prove that the feasible point of MPVCC is a local minimizer,it is also an AM-stationary point.It states that AM-stationarity concept is legitimate optimality condition.It is useful in the analysis of algorithms.Secondly,we know a way to measure the quality of a sequential optimality condition is relating to exact stationarity.We want to know under which MPVCC constraint qualification our sequential conditions guarantee M-stationary point.We present a new constraint qualification for M-stationarity,called AM-regularity,which can be used in the convergence analysis.The relation between old and new constraint qualifications are discussed.Finally,we consider an interior-penalty method to solve MPVCC,and show that the accumulation point of the sequence generated by the algorithm I is M-stationary point under AM-regularity constraint qualification.Numerical result indicates that our proposed algorithm is a capable strategy for MPVCC.