Piecewise M-Stationarity and Related Algorithms for Mathematical Programs with Complementarity Constraints

Abstract: This study explores B-stationarity of mathematical programs with complementarity constraints (MPCCs) and convergence behavior of MPCC algorithms. Special attention is given to the cases with biactive complementarity constraints. First, we propose the concept of piecewise M-stationarity and prove its equivalence to B-stationarity under MPCC-ACQ. Then, we investigate convergence properties of the NCP-based bounding methods we proposed in [31], without requiring MPCC-LICQ; an interpretation of the algorithm's behavior together with the concept of piecewise M-stationarity leads to a cost reduction in B-stationarity verification. In addition, practical issues related to convergence to non-strongly stationary solutions are discussed, which shows that the NCP-based complementarity reformulations have an advantage in avoiding unbounded multipliers near these solutions.