Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation

Shapiro, Alexander; Xu, Huifu

doi:10.1080/02331930801954177

Cited by 131 publications

(112 citation statements)

References 24 publications

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“…Robinson 1996;Gürkan et al 1999;Shapiro & Xu 2005;Shapiro 2006; and references therein). Using K random samples of the random vector u, a problem of the form (SMPEC), with p k Z 1=K for every k, is solved, for increased values of K. Shapiro & Xu (2005) study the convergence of this method for (SMPCC) U under conditions that are fulfilled if yð$; uÞ is unique and continuous for almost any u, X is non-empty and compact, and the function f ðx; yðx; $Þ; $Þ is bounded over X almost surely. Under additional conditions on S and f such that the expected value function is differentiable, it is also established that stationary points to discretizations converge to the set of stationary points to (SMPCC) U .…”

Section: (B ) Discretization Approachesmentioning

confidence: 99%

“…For SMPEC models, such techniques are known as the sample path method, sample average approximation, stochastic counterpart method and the simulated likelihood method (e.g. Robinson 1996;Gürkan et al 1999;Shapiro & Xu 2005;Shapiro 2006; and references therein). Using K random samples of the random vector u, a problem of the form (SMPEC), with p k Z 1=K for every k, is solved, for increased values of K. Shapiro & Xu (2005) study the convergence of this method for (SMPCC) U under conditions that are fulfilled if yð$; uÞ is unique and continuous for almost any u, X is non-empty and compact, and the function f ðx; yðx; $Þ; $Þ is bounded over X almost surely.…”

Section: (B ) Discretization Approachesmentioning

confidence: 99%

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Robust bi-level optimization models in transportation science

Patriksson

2008

Phil. Trans. R. Soc. A.

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Mathematical programmes with equilibrium constraints (MPECs) constitute important modelling tools for network flow problems, as they place 'what-if' analyses in a proper mathematical framework. We consider a class of stochastic MPEC traffic models that explicitly incorporate possible uncertainties in travel costs and demands. In stochastic programming terminology, we consider 'here-and-now' models where decisions must be made before observing the uncertain parameter values and the responses of the network users; the objective is to minimize the expectation of the upper-level objective function. Such a model could, for example, be used to derive a fixed toll pricing scheme that provides the best revenue for a given network over a time period, where variations in traffic conditions and demand elasticities are described by distributions of parameters in the travel time and demand functions.We present new results on the stability of globally optimal solutions to perturbations in the probability distribution, establishing the robustness of the model. We also discuss penalization and discretization algorithms, the latter enabling the use of standard MPEC algorithms, and provide many future research avenues.

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Section: (B ) Discretization Approachesmentioning

confidence: 99%

Section: (B ) Discretization Approachesmentioning

confidence: 99%

Robust bi-level optimization models in transportation science

Patriksson

2008

Phil. Trans. R. Soc. A.

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“…From computational perspective, one often need estimate the sample size N given a prescribed error bound d(x N , X * ). A popular way to address this issue is to consider the socalled exponential convergence, that is, with probability approaching one exponentially fast, {x N } converges to X * based on the classical Cramér's large deviation theorem [9], see [28,29] and the references therein.…”

Section: Convergence Analysismentioning

confidence: 99%

Monte Carlo methods for mean-risk optimization and portfolio selection

Zhang

2010

Comput Manag Sci

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Stochastic programming is a well-known instrument to model many risk management problems in finance. In this paper we consider a stochastic programming model where the objective function is the variance of a random function and the constraint function is the expected value of the random function. Instead of using popular scenario tree methods, we apply the well-known sample average approximation (SAA) method to solve it. An advantage of SAA is that it can be implemented without knowing the distribution of the random data. We investigate the asymptotic properties of statistical estimators obtained from the SAA problem including examining the rate of convergence of optimal solutions of the SAA problem as sample size increases. By using the classical penalty function technique and recent results on uniform exponential convergence of sample average random functions, we show that under some mild conditions the statistical estimator of the optimal solution converges to its true counterpart at an exponential rate. We apply the proposed model and the numerical method to a portfolio management problem and present some numerical results.

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“…In [9], the authors proposed a smoothing implicit programming approach for solving the SMPECs with a finite sample space. Subsequently, there have been a number of attempts [2,10,11,16,17,19] to deal with various models of SMPECs. In particular, Lin and Fukushima [10,11] suggested a smoothing penalty method and a regularization method, respectively, for a special class of here-and-now problems.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Lin and Fukushima [10,11] suggested a smoothing penalty method and a regularization method, respectively, for a special class of here-and-now problems. Shapiro and Xu [16,17,19] discussed the sample average approximation and implicit programming approaches for the lower-level wait-and-see problems. In addition, Birbil et al [2] considered an SMPEC in which both the objective and constraints involve expectations.…”

Section: Introductionmentioning

confidence: 99%

Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization

2007

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In this paper, we consider the stochastic mathematical programs with equilibrium constraints, which includes two kinds of models called here-and-now and lower-level wait-andsee problems. We present a combined smoothing implicit programming and penalty method for the problems with a finite sample space. Then, we suggest a quasi-Monte 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 so-called picnic vender decision problem.

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Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation

Cited by 131 publications

References 24 publications

Robust bi-level optimization models in transportation science

Robust bi-level optimization models in transportation science

Monte Carlo methods for mean-risk optimization and portfolio selection

Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization

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