A Probabilistic Lower Bound for Two-Stage Stochastic Programs

Dantzig, George B.; Infanger, Gerd

doi:10.1007/978-1-4419-1642-6_2

Cited by 5 publications

(11 citation statements)

References 32 publications

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“…Higle and Sen [21] have also proposed a statistical lower bound that is rooted in duality. Dantzig and Glynn [11], Dantzig and Infanger [12], Infanger [26,27], and Higle and Sen [19,23] use Monte Carlo versions of lower bounds obtained in sampling-based adaptations of deterministic cutting-plane algorithms. Throughout we focus on problems in which the feasible region, X, is simple in that feasibility of a candidate solution is easy to enforce and verify.…”

Section: Letmentioning

confidence: 99%

“…Consequently, the quality statement regardingx T is tighter when variability is low and larger when variability is high. When the sample sizes are increased according to (12) and the procedure stops at iteration T according to (11), the CI (13) is an asymptotically valid CI provided D n (x) satisfies a finite moment generating function (MGF) assumption. Next, we state the MGF assumption, and then we summarize this result along with the fact that sequential sampling stops in a finite number of iterations, w.p.1.…”

Section: Assumption 4 (A4mentioning

confidence: 99%

“…Let > > 0 and h > h > 0 and 0 < α < 1. Consider the above sequential sampling procedure where the sample size is increased according to (12), and the procedure stops at iteration T according to (11).…”

Section: Assumption 6 (A6) For Somementioning

confidence: 99%

“…Theorem 4 can be recovered when the MGF assumption is relaxed, at the price of a faster rate of growth in the sample size. When we assume that the MGF exists through (A6), the sample size growth is sublinear with respect to the iteration number k, of order O(log 2 k) in (12). When only the first two moments are finite, the growth in sample size is essentially linear, O(k).…”

Section: Assumption 6 (A6) For Somementioning

confidence: 99%

“…In (12), intelligent choices of the parameter q can help minimize the number of required samples and hence minimize the computational burden. This is especially important in applications where obtaining a new sample is costly and/or solving optimization problems with larger sample sizes become computationally prohibitive.…”

Section: Assumption 6 (A6) For Somementioning

confidence: 99%

See 4 more Smart Citations

Assessing Solution Quality in Stochastic Programs via Sampling

Bayraksan¹,

Morton²

2009

Decision Technologies and Applications

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Determining if a solution is optimal or near optimal is fundamental in optimization theory, algorithms, and computation. For instance, Karush-Kuhn-Tucker conditions provide necessary and sufficient optimality conditions for certain classes of problems, and bounds on optimality gaps are frequently used as part of optimization algorithms. Such bounds are obtained through Lagrangian, integrality, or semidefinite programming relaxations. An alternative approach in stochastic programming is to use Monte Carlo sampling-based estimators on the optimality gap. In this tutorial, we present a simple, easily implemented procedure that forms a point and interval estimator on the optimality gap of a given candidate solution. We then discuss methods to reduce the computational effort, bias, and variance of our simplest estimator. We also provide a framework that allows the use these optimality gap estimators in an algorithmic way by providing rules to iteratively increase the sample sizes and to terminate. This scheme can be used as a stand-alone sequential sampling procedure, or it can be used in conjunction with a variety of sampling-based algorithms to obtain a solution to a stochastic program with a priori control on the quality of that solution.

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Section: Letmentioning

confidence: 99%

Section: Assumption 4 (A4mentioning

confidence: 99%

Section: Assumption 6 (A6) For Somementioning

confidence: 99%

Section: Assumption 6 (A6) For Somementioning

confidence: 99%

Section: Assumption 6 (A6) For Somementioning

confidence: 99%

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Assessing Solution Quality in Stochastic Programs via Sampling

Bayraksan¹,

Morton²

2009

Decision Technologies and Applications

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Product-line planning under uncertainty

Karakaya

Köksal

2022

Computers & Operations Research

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pagesThis study addresses the problem of multi-period mix of product-lines under a product-family, which incorporates launching decisions of new products, capacity expansion decisions and product interdependencies. The problem is modelled as a two-stage stochastic program with recourse in which price, demand, production cost and cannibalisation effect of new products are treated as uncertain parameters.The solution approach employs the Sample Average Approximation based on Monte Carlo bounding technique and multi-cut version of L-shaped method to solve approximate problems efficiently, which is tested on different cases considering VSS and EVPI performance measures. The data collected through two experimental studies is analysed using ANOVA and Random Forest methodology in order to understand which problem parameters are significant on the performance measures and to generate some rule-based inferences reflecting the relationship between significant parameters and the performance of the proposed stochastic model.

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Fixed-Width Sequential Stopping Rules for a Class of Stochastic Programs

Bayraksan¹,

Pierre-Louis²

2012

SIAM J. Optim.

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Monte Carlo sampling-based methods are frequently used in stochastic programming when exact solution is not possible. A critical component of Monte Carlo sampling-based methods is determining when to stop sampling to ensure the desired quality of the solutions. In this paper, we develop stopping rules for sequential sampling procedures that depend on the width of an optimality gap confidence interval estimator. The procedures solve a sequence of sampling approximations with increasing sample size to generate solutions and stop when the width of the confidence interval on the current solution's optimality gap plus an inflation factor fall below a prespecified value,. We first present a method that takes the schedule of sample sizes as an input and provide guidelines on the growth of sample sizes. Then, we present a method that increases the sample sizes according to the current estimates of the optimality gap. The larger the estimates, the larger the increases in sample sizes. We provide conditions under which the procedures find-optimal solutions and terminate in a finite number of iterations with probability one and present empirical performance on test problems.

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A Probabilistic Lower Bound for Two-Stage Stochastic Programs

Cited by 5 publications

References 32 publications

Assessing Solution Quality in Stochastic Programs via Sampling

Assessing Solution Quality in Stochastic Programs via Sampling

Product-line planning under uncertainty

Fixed-Width Sequential Stopping Rules for a Class of Stochastic Programs

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