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Stochastic programming with equilibrium constraints
 Journal Optimization Theory and Applications
"... In this paper we discuss hereandnow type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We also discuss consistency and rates of conver ..."
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In this paper we discuss hereandnow type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We also discuss consistency and rates of convergence of sample average approximations of such stochastic problems. Key words: equilibrium constraints, twostage stochastic programming, variational inequalities, complementarity conditions, statistical inference, exponential rates. 1 1
Mathematical programs with vanishing constraints: Optimality conditions, sensitivity, and a relaxation method
 J. Optim. Theory Appl
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Sensitivity analysis for twolevel value functions with applications to bilevel programming
, 2011
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Mathematical programs with complementarity constraints: Convergence properties of a smoothing method
 Mathematics of Operations Research
, 2007
"... Abstract. In the present paper, optimization problems P with complementarity constraints are considered. Characterizations for local minimizers x of P of order one and two are presented. We analyze a parametric smoothing approach for solving these programs in which P is replaced by a perturbed probl ..."
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Abstract. In the present paper, optimization problems P with complementarity constraints are considered. Characterizations for local minimizers x of P of order one and two are presented. We analyze a parametric smoothing approach for solving these programs in which P is replaced by a perturbed problem P fi depending on a (small) parameter fi. We are interested in the convergence behavior of the feasible set F fi and the convergence of the solutions x fi of P fi for fi → 0. In particular, it is shown, that under generic assumptions the solutions x fi are unique and converge to a solution x of P with a rate O. fi/. Moreover, the convergence for the Hausdorff distance d.F fi;F / between the feasible sets of P fi and P is of order O. fi/.
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.
J Optim Theory Appl (2009) 142: 501–532 DOI 10.1007/s1095700995174 Mathematical Programs with Vanishing Constraints: Optimality Conditions, Sensitivity, and a Relaxation Method
, 2009
"... Abstract We consider a class of optimization problems with switchoff/switchon constraints, which is a relatively new problem model. The specificity of this model is that it contains constraints that are being imposed (switched on) at some points of the feasible region, while being disregarded (swi ..."
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Abstract We consider a class of optimization problems with switchoff/switchon constraints, which is a relatively new problem model. The specificity of this model is that it contains constraints that are being imposed (switched on) at some points of the feasible region, while being disregarded (switched off) at other points. This seems to be a potentially useful modeling paradigm, that has been shown to be helpful, for example, in optimal topology design. The fact that some constraints “vanish ” from the problem at certain points, gave rise to the name of mathematical programs with vanishing constraints (MPVC). It turns out that such problems are usually degenerate at a solution, but are structurally different from the related class of mathematical programs with complementarity constraints (MPCC). In this paper, we first discuss some known first and secondorder necessary optimality conditions for MPVC, giving new very short and direct justifications. We then derive some new special secondorder sufficient optimality conditions for these problems and show that, quite remarkably, these conditions are actually equivalent to the classical/standard secondorder sufficient conditions in optimization. We also provide a sensitivity analysis for MPVC. Communicated by B.T. Polyak.