Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints ✩

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations.We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow–Hurwicz–Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn–Tucker constraint qualification, and MPEC Abadie constraint qualification.  2004 Elsevier Inc. All rights reserved.