Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints
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Second-Order Optimality Conditions for Mathematical Programs with Equilibrium Constraints Lei Guo · Gui-Hua Lin · Jane J. Ye
Received: 27 June 2012 / Accepted: 15 November 2012 / Published online: 4 December 2012 © Springer Science+Business Media New York 2012
Abstract We study second-order optimality conditions for mathematical programs with equilibrium constraints (MPEC). Firstly, we improve some second-order optimality conditions for standard nonlinear programming problems using some newly discovered constraint qualifications in the literature, and apply them to MPEC. Then, we introduce some MPEC variants of these new constraint qualifications, which are all weaker than the MPEC linear independence constraint qualification, and derive several second-order optimality conditions for MPEC under the new MPEC constraint qualifications. Finally, we discuss the isolatedness of local minimizers for MPEC under very weak conditions. Keywords Mathematical program with equilibrium constraints · Second-order optimality condition · Constraint qualification · Isolatedness
1 Introduction MPEC is a constrained optimization problem in which the essential constraints are defined by some parametric variational inequalities or parametric complementarity
Communicated by Michael Patriksson. L. Guo School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China e-mail: [email protected] G.-H. Lin () School of Management, Shanghai University, Shanghai 200444, China e-mail: [email protected] J.J. Ye Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3P4, Canada e-mail: [email protected]
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J Optim Theory Appl (2013) 158:33–64
systems. It plays a very important role in many fields, such as engineering design, economic equilibria, transportation science, multilevel game, and mathematical programming itself. However, this kind of problems is generally difficult to deal with because its constraints fail to satisfy the standard Mangasarian–Fromovitz constraint qualification (MFCQ) at any feasible point [1]. A lot of research has been done during the last two decades to study the first-order optimality conditions for MPEC, such as Clarke (C-), Mordukhovich (M-), Strong (S-), Bouligrand (B-) stationarity conditions; see, e.g., [1–8]. At the same time, algorithms for solving MPEC have been proposed by using a number of approaches, such as sequential quadratic programming approach, penalty function approach, relaxation approach, active set identification approach, etc.; see, e.g., [9–11] and the references therein for more details. In this paper, we focus on second-order optimality conditions for MPEC. Firstorder optimality conditions tell us how the first derivatives of the functions involved are related to each other at locally optimal solutions. However, for some feasible directions in the tangent cone such as the so-called critical directions, we cannot determine from the first derivative information alone whether the objective function increases or decreases in this directi
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