A Gap Penalty Reformulation for Mathematical Programming with Complementarity Constraints: Convergence Analysis

Abstract: Our recent study (Lin and Ohtsuka, 2024) proposed a new penalty method for solving mathematical programming with complementarity constraints (MPCC). This method first reformulates MPCC as a parameterized nonlinear programming called gap penalty reformulation and then solves a sequence of gap penalty reformulations with an increasing penalty parameter. This study examines the convergence behavior of the new penalty method. We prove that it converges to a strongly stationary point of MPCC, provided that: (i) The MPCC linear independence constraint qualification holds. (ii) The upper-level strict complementarity condition holds. (iii) The gap penalty reformulation satisfies the second-order necessary conditions in terms of the second-order directional derivative. Because strong stationarity is used to identify the MPCC local minimum, our analysis indicates that the new penalty method can find an MPCC solution.