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

TL;DR

This paper introduces the concept of piecewise M-stationarity for MPCCs, demonstrating its equivalence to B-stationarity and analyzing convergence of NCP-based algorithms without LICQ, highlighting practical advantages.

Contribution

It proposes piecewise M-stationarity, proves its equivalence to B-stationarity under MPCC-ACQ, and analyzes convergence of NCP-based methods without LICQ.

Findings

Piecewise M-stationarity is equivalent to B-stationarity under MPCC-ACQ.

NCP-based algorithms converge without requiring MPCC-LICQ.

Reformulations help avoid unbounded multipliers near non-strongly stationary solutions.

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.