Sensitivity Analysis of the Value Function for Parametric Mathematical Programs with Equilibrium Constraints

In this paper, we perform sensitivity analysis of the value function for parametric mathematical programs with equilibrium constraints (MPEC). We show that the value function is directionally differentiable in every direction under the MPEC relaxed constant rank regularity condition, the MPEC no nonzero abnormal multiplier constraint qualification, and the restricted inf-compactness condition. This result is new even in the setting of nonlinear programs in which case it means that under the relaxed constant rank regularity condition, the Mangasarian–Fromovitz constraint qualification, and the restricted inf-compactness condition, the value function for parametric nonlinear programs is directionally differentiable in every direction. Enhanced Mordukhovich (M-) and Clarke (C-) stationarity conditions are Mand C-stationarity conditions with certain enhanced properties and the sets of enhanced Mand C-multipliers are usually smaller than their associated sets of Mand C-multipliers. In this paper, we give upper estimates for the subdifferential of the value function in terms of the enhanced Mand C-multipliers, respectively. Such estimates give sharper results than their Mand C-counterparts.