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SENSITIVITY ANALYSIS OF THE VALUE FUNCTION FOR PARAMETRIC MATHEMATICAL PROGRAMS WITH EQUILIBRIUM CONSTRAINTS
, 2014
"... 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 n ..."
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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 infcompactness 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 infcompactness condition, the value function for parametric nonlinear programs is directionally differentiable in every direction. Enhanced Mordukhovich (M) and Clarke (C) stationarity conditions are M and Cstationarity conditions with certain enhanced properties and the sets of enhanced M and Cmultipliers are usually smaller than their associated sets of M and Cmultipliers. In this paper, we give upper estimates for the subdifferential of the value function in terms of the enhanced M and Cmultipliers, respectively. Such estimates give sharper results than their M and Ccounterparts.
Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraints
"... In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the MangasarianFromovitz condition and ..."
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In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the MangasarianFromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudoLipschitz continuity and calmness of certain setvalued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.
Necessary Optimality Conditions for Optimal Control Problems with Equilibrium Constraints
, 2015
"... Abstract. This paper introduces and studies the optimal control problem with equilibrium constraints (OCPEC). The OCPEC is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint and it can be seen as a dynamic mathematical program ..."
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Abstract. This paper introduces and studies the optimal control problem with equilibrium constraints (OCPEC). The OCPEC is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint and it can be seen as a dynamic mathematical program with equilibrium constraints (MPEC). It provides a powerful modeling paradigm for many practical problems such as bilevel optimal control problems and dynamic principalagent problems. In this paper, we propose Fritz John type weak, Clarke, Mordukhovich and strong stationarities for the OCPEC. Moreover, we give some sufficient conditions to ensure that the local minimizers of the OCPEC are Fritz John type weakly stationary, Mordukhovich stationary and strongly stationary respectively. Key Words. Optimal control problem with equilibrium constraints, necessary optimality condition, weak stationarity, Mordukhovich stationarity, strong stationarity.