A New Relaxed Method For Mathematical Programs With Complementarity Constraints

In this thesis, mathematical programs with complementarity constraints (MPCC for short) which are a special class of constrained optimization are investigated. MPCC plays an important role in many fields, such as econo-my, traffic transportation, network design, engineering design and so on. The significance of the studies on MPCC is therefore evident both in theory and application. Because of the complementarity constraints, most of the con-straint qualifications of standard nonlinear programming are violated at any feasible point of MPCC. Hence, the effective algorithms of standard nonlin-ear programming can not be employed directly to solve MPCC.In this thesis a new relaxed method for MPCC is proposed. Firstly, based on Mangasarian complementarity function, MPCC is relaxed. The relaxed problem is a parametrized nonlinear programming. Secondly, it is proved that the sequence of stationary points of the relaxed problems converges to M-stationary point of MPCC under some mild assumptions, such as constant positive linear dependence (MPCC-CPLD for short); further, it is shown that the stationary point is strong for MPCC if some additional conditions hold-s. Thirdly, we analyze the existence of multipliers for the relaxed problem. We show that Guignard constraint qualification holds for the relaxed problem under MPCC-linear independence constraint qualifications (MPCC-LICQ for short), and then obtain the existence theorem of the Lagrange multipliers. Fi-nally, numerical experiments are implemented, and the preliminary numerical results show that the proposed method is feasible and effective.