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Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization
"... Abstract In this paper, we consider the stochastic mathematical programs with equilibrium constraints, which includes two kinds of models called hereandnow and lowerlevel waitandsee problems. We present a combined smoothing implicit programming and penalty method for the problems with a finite ..."
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Abstract In this paper, we consider the stochastic mathematical programs with equilibrium constraints, which includes two kinds of models called hereandnow and lowerlevel waitandsee problems. We present a combined smoothing implicit programming and penalty method for the problems with a finite sample space. Then, we suggest a quasiMonte Carlo approximation method for solving a problem with continuous random variables. A comprehensive convergence theory is included as well. We further report numerical results with the socalled picnic vender decision problem. Key words. Stochastic mathematical program with equilibrium constraints, waitandsee, hereandnow, smoothing implicit programming, penalty method, quasiMonte Carlo method. 1
S.: Sample average approximation of expected value constrained stochastic programs, Eprint available at: http://www.optimizationonline.org
, 2007
"... We propose a sample average approximation (SAA) method for stochastic programming problems involving an expected value constraint. Such problems arise, for example, in portfolio selection with constraints on conditional valueatrisk (CVaR). Our contributions include an analysis of the convergence ..."
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Cited by 15 (0 self)
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We propose a sample average approximation (SAA) method for stochastic programming problems involving an expected value constraint. Such problems arise, for example, in portfolio selection with constraints on conditional valueatrisk (CVaR). Our contributions include an analysis of the convergence rate and a statistical validation scheme for the proposed SAA method. Computational results using a portfolio selection problem with a CVaR constraint are presented. Key words: Sample average approximation; Expected value constrained stochastic program; Conditional valueatrisk; Convergence rate; Validation scheme; Portfolio optimization
Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints
, 2007
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Monte Carlo and QuasiMonte Carlo Sampling Methods for a Class of Stochastic Mathematical Programs with Equilibrium Constraints
, 2005
"... In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (2004). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with ..."
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Cited by 9 (5 self)
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In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (2004). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasiMonte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasiMonte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this example.
Stochastic Equilibrium Problems and Stochastic Mathematical Programs with Equilibrium Constraints: A Survey
, 2009
"... In the recent optimization research community, various equilibrium problems and related problems under uncertainty have drawn increasing attention. Novel formulations and numerical methods have been proposed to deal with those problems. This paper provides a brief review of the recent developments ..."
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Cited by 9 (3 self)
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In the recent optimization research community, various equilibrium problems and related problems under uncertainty have drawn increasing attention. Novel formulations and numerical methods have been proposed to deal with those problems. This paper provides a brief review of the recent developments in the topics including stochastic variational inequality problems, stochastic complementarity problems and stochastic mathematical programs with equilibrium constraints.
D.: Stochastic Nash equilibrium problems: sample average approximation and applications. Available at http://www.optimizationonline.org
, 2009
"... Abstract This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The wellknown sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke ge ..."
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Cited by 7 (3 self)
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Abstract This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The wellknown sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a statistical estimator (a Nash equilibrium or a NashCstationary point) obtained from sample average approximate equilibrium problem converges to its true counterpart. Moreover, under some calmness conditions of the Clarke generalized derivatives, it is shown that with probability approaching one exponentially fast by increasing sample size, the NashCstationary point converges to a weak NashCstationary point of the true problem. Finally, the model is applied to stochastic Nash equilibrium problem in the wholesale electricity market.
Stability analysis of two stage stochastic mathematical programs with complementarity constraints via NLPregularization
, 2010
"... This paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint: first, the complementarity constraint is approximated by a parameterized system of inequalities with a wellknown regu ..."
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Cited by 6 (6 self)
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This paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint: first, the complementarity constraint is approximated by a parameterized system of inequalities with a wellknown regularization approach [44] in deterministic mathematical programs with equilibrium constraints; the distribution of the random variables of the regularized two stage stochastic program is then approximated by a sequence of probability measures. By treating the approximation problems as a perturbation of the original (true) problem, we carry out a detailed stability analysis of the approximated problems including continuity and local Lipschitz continuity of optimal value functions, and outer semicontinuity and continuity of the set of optimal solutions and stationary points. A particular focus is given to the case when the probability distribution is approximated by the empirical probability measure which is known as sample average approximation.
New reformulations for stochastic nonlinear complementarity problems
 Optimization Methods and Software
"... We consider the stochastic nonlinear complementarity problem (SNCP). We first formulate the problem as a stochastic mathematical program with equilibrium constraints and then, in order to develop efficient algorithms, we give some reformulations of the problem. Furthermore, based on the reformulatio ..."
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Cited by 6 (2 self)
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We consider the stochastic nonlinear complementarity problem (SNCP). We first formulate the problem as a stochastic mathematical program with equilibrium constraints and then, in order to develop efficient algorithms, we give some reformulations of the problem. Furthermore, based on the reformulations, we propose a smoothed penalty method for solving SNCP. A rigorous convergence analysis is also given. Key Words. stochastic nonlinear complementarity problem, stochastic mathematical program with equilibrium constraints, stationarity, subdifferential, convergence.
New Restricted NCP Functions and Their Applications to Stochastic NCP and Stochastic MPEC
, 2006
"... We focus on studying stochastic nonlinear complementarity problems (SNCP) and stochastic mathematical programs with equilibrium constraints (SMPEC). Instead of the NCP functions employed in the literature, we use the restricted NCP functions to define expected residual minimization formulations for ..."
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Cited by 6 (4 self)
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We focus on studying stochastic nonlinear complementarity problems (SNCP) and stochastic mathematical programs with equilibrium constraints (SMPEC). Instead of the NCP functions employed in the literature, we use the restricted NCP functions to define expected residual minimization formulations for SNCP and SMPEC. We then discuss level set conditions and error bounds of the new formulation. Numerical examples show that the new formulations have some desirable properties which the existing ones do not have.
Quantitative Stability Analysis of Stochastic Generalized Equations
, 2012
"... We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random setvalued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stocha ..."
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Cited by 4 (2 self)
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We consider the solution of a system of stochastic generalized equations (SGE) where the underlying functions are mathematical expectation of random setvalued mappings. SGE has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem and a stochastic equilibrium problem. We derive quantitative continuity of expected value of the setvalued mapping with respect to the variation of the underlying probability measure in a metric space. This leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the SGE. Under some metric regularity conditions, we derive Aubin’s property of the solution set mapping with respect to the change of probability measure. The established results are applied to stability analysis of stationary points of classical one stage and two stage stochastic minimization problems, two stage stochastic mathematical programs with equilibrium constraints and stochastic programs with second order dominance constraints. Key words. Stochastic generalized equations, stability analysis, equicontinuity, one stage stochastic programs, two stage stochastic programs, two stage SMPECs, stochastic semiinfinite programming 1