A branch and bound method for stochastic global optimization

Norkin, V. I.; Pflug, Georg Ch.; Ruszczyński, Andrzej

doi:10.1007/bf02680569

Cited by 229 publications

(138 citation statements)

References 9 publications

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“…Perceived Cost Estimates As indicated by the inequalities in (1.1), and previously shown by Norkin et al (1998) and Mak et al (1999), the expected perceived cost:…”

Section: Methodsmentioning

confidence: 66%

The impact of sampling methods on bias and variance in stochastic linear programs

Freimer¹,

Linderoth

Thomas

2010

Comput Optim Appl

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Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the problem based on this subset, an approach known as sample path optimization. The value of the optimal solution to the sampled problem provides an estimate of the true objective function value. This estimator is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative sampling methods, antithetic variates and Latin Hypercube sampling, affect both the bias and variance, and thus the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. Whether the bias reduction or variance reduction plays a larger role in MSE reduction is problem and parameter specific.

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“…Perceived Cost Estimates As indicated by the inequalities in (1.1), and previously shown by Norkin et al (1998) and Mak et al (1999), the expected perceived cost:…”

Section: Methodsmentioning

confidence: 66%

The impact of sampling methods on bias and variance in stochastic linear programs

Freimer¹,

Linderoth

Thomas

2010

Comput Optim Appl

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“…Table 5 shows a decrease in both these terms as the batch size m grows. In fact, it is possible to show that EU m decreases monotonically in m [17,14]. The increase in CPU times with larger batch sizes in Table 5 (and to a lesser extent in Table 4) is due, in part, to the IP (15) becoming larger.…”

Section: Procedures Mcskpmentioning

confidence: 85%

On a Stochastic Knapsack Problem and Generalizations

Morton

Wood

1998

Operations Research/Computer Science Interfaces Series

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Abstract:We consider an integer stochastic knapsack problem (SKP) where the weight of each item is deterministic, but the vector of returns for the items is random with known distribution. The objective is to maximize the probability that a total return threshold is met or exceeded. We study several solution approaches. Exact procedures, based on dynamic programming (DP) and integer programming (IP), are developed for returns that are independent normal random variables with integral means and variances. Computation indicates that the DP is significantly faster the most efficient algorithm to date. The IP is less efficient, but is applicable to more general stochastic IPs with independent normal returns. We also develop a Monte Carlo approximation procedure to solve SKPs with general distributions on the random returns. This method utilizes upper-and lower-bound estimators on the true optimal solution value 149 150 in order to construct a confidence interval on the optimality gap of a candidate solution.

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“…A discussion of two-stage stochastic integer programs with recourse can be found in Birge and Louveaux (1997). A branch and bound approach for solving stochastic discrete optimization problems was suggested by Norkin, Pflug and Ruszczyński (1998). Schultz, Stougie and Van der Vlerk (1998) …”

Section: Stochastic Programming With Recoursementioning

confidence: 99%

Stochastic Optimization

2006

Scientific Computation

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A branch and bound method for stochastic global optimization

Cited by 229 publications

References 9 publications

The impact of sampling methods on bias and variance in stochastic linear programs

The impact of sampling methods on bias and variance in stochastic linear programs

On a Stochastic Knapsack Problem and Generalizations

Stochastic Optimization

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