Convergence Properties of Two-Stage Stochastic Programming

Dai, Liyi; Chen, C. H.; Birge, John R.

doi:10.1023/a:1004649211111

Cited by 82 publications

(65 citation statements)

References 12 publications

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“…In this section we show that the probability of getting an optimal solution this way approaches 1 exponentially fast as we increase the sample size. We use large deviations theory and a result due to Dai, Chen, and Birge [2000] to prove our statement. Vogel [1988] shows, under weaker conditions, that the feasible region of a sample average approximation for a chance constraint problem approaches the true feasible region at a polynomial rate and conjectures, without giving a proof, that an exponential rate of convergence is attainable under similar conditions to those we impose.…”

Section: Exponential Rate Of Convergence Of Optimal Solutions Of the mentioning

confidence: 87%

“…We apply the strong law of large numbers to prove conditions for almost sure convergence and apply a result due to Dai, Chen, and Birge [2000] to prove an exponential rate of convergence of the optimal solutions as the sample size increases. [Vogel, 1994] proved almost sure convergence in a similar setting, but we include proofs for reasons listed in Section 4.1.…”

mentioning

confidence: 99%

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Call Center Staffing with Simulation and Cutting Plane Methods

Atlason

Epelman

Henderson

2004

Annals of Operations Research

141

117

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We present an iterative cutting plane method for minimizing staffing costs in a service system subject to satisfying acceptable service level requirements over multiple time periods. We assume that the service level cannot be easily computed, and instead is evaluated using simulation. The simulation uses the method of common random numbers, so that the same sequence of random phenomena is observed when evaluating different staffing plans. In other words, we solve a sample average approximation problem.We establish convergence of the cutting plane method on a given sample average approximation. We also establish both convergence, and the rate of convergence, of the solutions to the sample average approximation to solutions of the original problem as the sample size increases. The cutting plane method relies on the service level functions being concave in the number of servers. We show how to verify this requirement as our algorithm proceeds. A numerical example showcases the properties of our method, and sheds light on when the concavity requirement can be expected to hold.

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Section: Exponential Rate Of Convergence Of Optimal Solutions Of the mentioning

confidence: 87%

mentioning

confidence: 99%

Call Center Staffing with Simulation and Cutting Plane Methods

Atlason

Epelman

Henderson

2004

Annals of Operations Research

141

117

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show abstract

“…See Dai et al (2000), Parpas et al (2013), and Homem-de-Mello and Bayraksan (2014) for the two-stage stochastic programming setup of the problem. Suppose that a newsvendor can choose the number of newspapers, x, to be purchased at the first stage.…”

Section: Newsvendor Problemmentioning

confidence: 99%

“…Shapiro (2003) and Shapiro et al (2009) discuss the theoretical properties of sampling-based approaches comprehensively. Dai et al (2000), Shapiro and Homem-de-Mello (2000), and Polson and Sorensen (2011) also provide discussions and comparisons of the convergence rates of simulationbased optimization methods.…”

Section: Simulation-based Stochastic Programmingmentioning

confidence: 99%

Augmented Markov Chain Monte Carlo Simulation for Two-Stage Stochastic Programs with Recourse

2014

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I n this paper, we develop a simulation-based approach for two-stage stochastic programs with recourse.We construct an augmented probability model with stochastic shocks and decision variables. Simulating from the augmented probability model solves for the expected recourse function and the optimal first-stage decision. Markov chain Monte Carlo methods, together with ergodic averaging, provide a framework to compute the optimal solution. We illustrate our methodology via the two-stage newsvendor problem with unimodal and bimodal continuous uncertainty. Finally, we present performance comparisons of our algorithm and the sample average approximation method.

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“…This motivates one to study SAA under non-iid sampling. Dai, Chen and Birge [8] investigated the convergence of SAA estimators under general sampling (including iid and non-iid). They used the well-known Gärther-Ellis theorem [9] to establish the exponential convergence.…”

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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Convergence Properties of Two-Stage Stochastic Programming

Cited by 82 publications

References 12 publications

Call Center Staffing with Simulation and Cutting Plane Methods

Call Center Staffing with Simulation and Cutting Plane Methods

Augmented Markov Chain Monte Carlo Simulation for Two-Stage Stochastic Programs with Recourse

Monte Carlo methods for mean-risk optimization and portfolio selection

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