Simulation-based confidence bounds for two-stage stochastic programs

Infanger, Gerd

doi:10.1007/s10107-012-0621-0

Cited by 19 publications

(5 citation statements)

References 36 publications

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“…The commonly-investigated alternative sampling techniques for variance reduction in (SP) include (i) importance sampling [e.g., Barrera et al, 2016, Dantzig and Glynn, 1990, Higle, 1998, Infanger, 1992, Glynn and Infanger, 2013, Kozmík and Morton, 2015, (ii) (randomized) Quasi-Monte Carlo (QMC) [e.g., Pennanen and Koivu, 2005, Koivu, 2005, Homem-de-Mello, 2008, Homem-de-Mello et al 2011, Heitsch et al, 2016, (iii) AV [e.g., Freimer et al, 2012, Higle, 1998, Koivu, 2005, and (iv)…”

Section: Related Literaturementioning

confidence: 99%

Variance Reduction for Sequential Sampling in Stochastic Programming

Park,

Stockbridge,

Bayraksan

2020

Preprint

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This paper investigates the variance reduction techniques Antithetic Variates (AV) and Latin Hypercube Sampling (LHS) when used for sequential sampling in stochastic programming and presents a comparative computational study. It shows conditions under which the sequential sampling with AV and LHS satisfy finite stopping guarantees and are asymptotically valid, discussing LHS in detail. It computationally compares their use in both the sequential and nonsequential settings through a collection of two-stage stochastic linear programs with different characteristics. The numerical results show that while both AV and LHS can be preferable to random sampling in either setting, LHS typically dominates in the non-sequential setting while performing well sequentially and AV gains some advantages in the sequential setting. These results imply that, given the ease of implementation of these variance reduction techniques, armed with the same theoretical properties and improved empirical performance relative to random sampling, AV and LHS sequential procedures present attractive alternatives in practice for a class of stochastic programs.

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Section: Related Literaturementioning

confidence: 99%

Variance Reduction for Sequential Sampling in Stochastic Programming

Park,

Stockbridge,

Bayraksan

2020

Preprint

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“…From an algorithmic perspective, the stochastic decomposition framework initially developed by [31] is perhaps the most well-known practical approach that exploits the connections between statistical inference, sampling, and stochastic LPs. In addition, [28] proposes simulation-based Benders decomposition approach as a variant of the stochastic sub-gradient method specifically for 2SLPs and develops statistical confidence bounds for the optimal values.…”

Section: Introduction the Two-stage Stochastic Linear Program (2slpmentioning

confidence: 99%

Adaptive Sequential SAA for Solving Two-stage Stochastic Linear Programs

Pasupathy,

Song

2020

Preprint

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We present adaptive sequential SAA (sample average approximation) algorithms to solve large-scale two-stage stochastic linear programs. The iterative algorithm framework we propose is organized into outer and inner iterations as follows: during each outer iteration, a sample-path problem is implicitly generated using a sample of observations or "scenarios," and solved only imprecisely, to within a tolerance that is chosen adaptively, by balancing the estimated statistical error against solution error. The solutions from prior iterations serve as warm starts to aid efficient solution of the (piecewise linear convex) sample-path optimization problems generated on subsequent iterations. The generated scenarios can be independent and identically distributed (iid), or dependent, as in Monte Carlo generation using Latin-hypercube sampling, antithetic variates, or randomized quasi-Monte Carlo. We first characterize the almost-sure convergence (and convergence in mean) of the optimality gap and the distance of the generated stochastic iterates to the true solution set. We then characterize the corresponding iteration complexity and work complexity rates as a function of the sample size schedule, demonstrating that the best achievable work complexity rate is Monte Carlo canonical and analogous to the generic O(ǫ −2 ) optimal complexity for non-smooth convex optimization. We report extensive numerical tests that indicate favorable performance, due primarily to the use of a sequential framework with an optimal sample size schedule, and the use of warm starts. The proposed algorithm can be stopped in finite-time to return a solution endowed with a probabilistic guarantee on quality.

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“…Convexity is a key structural property that can be exploited in many ways. Assuming the minimum is attained, one can use gradient-based methods (for smooth functions) or a cutting-plane based method (for nonsmooth functions) to quickly find the minimum, or bounds on the minimum, e.g., Nesterov (2004), Glynn and Infanger (2013). Even if a function is not globally convex, one might use our methodology to identify regions around local minima in which the restriction of the objective function is convex ("basins of attraction").…”

Section: Introductionmentioning

confidence: 99%

Estimating the Probability that a Function Observed with Noise Is Convex

Jian

Henderson

2019

INFORMS Journal on Computing

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Consider a real-valued function that can only be observed with stochastic noise at a finite set of design points within a Euclidean space. We wish to determine whether there exists a convex function that goes through the true function values at the design points. We develop an asymptotically consistent Bayesian sequential sampling procedure that estimates the posterior probability of this being true. In each iteration, the posterior probability is estimated using Monte Carlo simulation. We offer three variance reduction methods -change of measure, acceptance-rejection, and conditional Monte Carlo. Numerical experiments suggest that the conditional Monte Carlo method should be preferred.

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Simulation-based confidence bounds for two-stage stochastic programs

Cited by 19 publications

References 36 publications

Variance Reduction for Sequential Sampling in Stochastic Programming

Variance Reduction for Sequential Sampling in Stochastic Programming

Adaptive Sequential SAA for Solving Two-stage Stochastic Linear Programs

Estimating the Probability that a Function Observed with Noise Is Convex

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