TAILIEUCHUNG - On sufficiency for mathematical programming problems with equilibrium constraints

In this paper, we consider a mathematical programming with equilibrium constraints (MPEC) where the objective and constraint functions are continuously differentiable. We establish the sufficient optimality condition for strict local minima of order m under the assumptions of generalized strong convexity of order m. | Yugoslav Journal of Operations Research 23 (2013) Number 2, 173-182 DOI: ON SUFFICIENCY FOR MATHEMATICAL PROGRAMMING PROBLEMS WITH EQUILIBRIUM CONSTRAINTS Shashi Kant MISHRA Yogendra PANDEY Department of Mathematics Faculty of Science Banaras Hindu University, Varanasi-221 005, India Received: February 2013 / Accepted: May 2013 Abstract: In this paper, we consider a mathematical programming with equilibrium constraints (MPEC) where the objective and constraint functions are continuously differentiable. We establish the sufficient optimality condition for strict local minima of order m under the assumptions of generalized strong convexity of order m. Keywords: Mathematical programming with equilibrium constraint, sufficient optimality condition, higher-order strong convexity. MSC: 90C30, 90C33, 90C46. 1. INRODUCTION Mathematical programming with equilibrium constraints (MPEC) is an optimization problem, where a parameter dependent variational inequality or, more specifically, a parameter dependent complementarity problem arises as a side constraint. MPEC is an extension of the class of bilevel programs, which was introduced in the operations research literature in the early 1970s by Bracken and McGill. So, MPECs are also called generalized bilevel programming problems. Book by Luo et al. [7] provides a solid foundation and an extensive study of MPEC. Many authors are developing interesting results on these topics, see for example [6, 9, 14, 15, 16]. The M-stationary condition in MPEC was first introduced in Ye and Ye [16] by using Mordukhovich 174 S. K. Mishra, Y. Pandey / On Sufficiency For Mathematical Programming coderivative of setvalued maps. Ye [13] established W-stationary, C-stationary, Astationary and S-stationary conditions for MPEC; he showed that the M-stationary condition is the most appropriate stationary condition for MPEC, in the sense that it is the second strongest stationary condition (with the .

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