Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity

Abstract

We study mathematical programs with complementarity constraints. Several stationarity concepts, based on a piecewise smooth formulation, are presented and compared. The concepts are related to stationarity conditions for certain smooth programs as well as to stationarity concepts for a nonsmooth exact penalty function. Further, we present Fiacco-McCormick type second order optimality conditions and an extension of the stability results of Robinson and Kojima to mathematical programs with complementarity constraints.

Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

“…Without the information on the sign of p * and μ * on the biactive set, the system (2.3)-(2.7) denotes C-stationarity, which is a weaker form of stationarity. For a survey of different stationarity concepts for finite-dimensional optimization problems with complementarity constraints we refer to [19].…”

“…Problem (P) can be viewed as multi-level optimization problem. While such problems have received repeated attention in the finite-dimensional context, see for instance [19], this point of view is not common in the infinite dimensional setting. See, however, the recent contribution [9].…”

An MPCC Formulation and Its Smooth Solution Algorithm for Continuous Network Design Problem

“…And, y 6 reaches the largest link capacity expansion when d=25, 30,35,40,45,50. The capacity improvement on link y 16 reaches the largest link capacity expansion when d=10, 15,30. In the third part of the experiment, the objective function only considers the total travel times, i.…”

“…The complementarity constraint [30] requires a product of two non-negative variables to be zero, consequently making their values complementary, i.e. when one variable is positive, the other must be zero.…”

“…Thus, the CNDP model can be formulated as mathematical programs with complementarity constraints (MPCC). However, solving MPCC is a hard task because the Mangasarian Fromovitz constraint qualification (MFCQ) does not hold at any feasible point [29,30]. To circumvent these problems, some algorithmic approaches have focused on avoiding this formulation.…”