| Mathemctical Programming with vertical complementarity constrained(MPVCC) optimization probrem is a new type of equilibrium model, which covers the classical nonlinear programming model and mathematical programming problems with complementarity constraints(MPCC) and is widely used in economic equilibrium, engineering, transportation, network design and many other areas. As most classical constraint qualifications in the general nonlinear programming problems do not hold for this kind of problem, Karush-Kuhn-Tucker(KKT) conditions may be not the first order optimality conditions, which makes the first order optimality conditions have many forms, such as M-stationary point, S-statinary point(strong statinary point) condition, W-stable point(weak statinary point) conditions. As we all know, constraint qualifications are basis of designing algorithms for MPVCC, thus seeking appropriate constraint qualification condition to ensure various stationary point condition for MPVCC has important theoretical significance and application value. This article focuses on the issue of constraint qualifications for MPVCC and applications. The specific contents are as follows:Firstly, after summarizing existing constraint qualifications for MPVCC, propose several new constraint qualifications, such as MPVCC constant rank constraint specification, MPVCC constant positive linear dependence constraint qualification and MPVCC relaxed constant positive linear dependence constraint qualification and study the relationship between them and existing constraint qualifications.Secondly, the new constraint qualifications for MPVCC are applied to the convergence theory of a class of relaxation method for solving MPVCC. Specifically, propose a relaxation method for solving MPVCC, and under the new constraint qualification establish the convergence theory.Finally, after constructing economic models based on game theory to a MPVCC model, solve this model by the relaxation method, numerical examples demonstrate the effectiveness of relaxation method. |
