The Augmented Lagrangian Method for Mathematical Programs with Vertical Complementarity Constraints Based on Inexact Scholtes Regularization

Mathematical program with vertical complementarity constraints (MPVCC) plays an important role in economics and engineering. Due to the vertical complementarity structure, most of the standard constraint qualifications fail to hold at a feasible point. Without constraint qualifications, the Karush–Kuhn–Tucker (KKT) conditions are not necessary optimality conditions. The classical methods for solving constrained optimization problems applied to MPVCC are likely to fail. It is necessary to establish efficient algorithms for solving MPVCC from both theoretical and numerical points of view. We present an algorithm to obtain stationarity of MPVCC by solving a sequence of Scholtes regularized problems. We consider the Scholtes regularization method with the sequence of approximate KKT points only. We prove that, under strictly weaker constraint qualifications, the accumulation point of the approximate KKT points is Clarke (C-) stationary point. In particular, we can get Mordukhovich (M-) or strongly (S-) stationary point under additional assumptions. From these results, we apply an augmented Lagrangian method to obtain a solution of MPVCC and give the convergence analysis. In particular, the accumulation point of the generated iterates is an S-stationary point if some boundedness conditions hold. The numerical results show that it is an effective way to solve MPVCC.