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57
The sample average approximation method for stochastic discrete optimization
 SIAM Journal on Optimization
, 2001
"... Abstract. In this paper we study a Monte Carlo simulation based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and consequently the expected value function is approximated by the corresponding sample average function. The ob ..."
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Cited by 146 (16 self)
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Abstract. In this paper we study a Monte Carlo simulation based approach to stochastic discrete optimization problems. The basic idea of such methods is that a random sample is generated and consequently the expected value function is approximated by the corresponding sample average function. The obtained sample average optimization problem is solved, and the procedure is repeated several times until a stopping criterion is satisfied. We discuss convergence rates and stopping rules of this procedure and present a numerical example of the stochastic knapsack problem. Key words. Stochastic programming, discrete optimization, Monte Carlo sampling, Law of Large Numbers, Large Deviations theory, sample average approximation, stopping rules, stochastic knapsack problem AMS subject classifications. 90C10, 90C15
The Empirical Behavior of Sampling Methods for Stochastic Programming
 Annals of Operations Research
, 2002
"... We investigate the quality of solutions obtained from sampleaverage approximations to twostage stochastic linear programs with recourse. We use a recently developed software tool executing on a computational grid to solve many large instances of these problems, allowing us to obtain highquality s ..."
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Cited by 78 (13 self)
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We investigate the quality of solutions obtained from sampleaverage approximations to twostage stochastic linear programs with recourse. We use a recently developed software tool executing on a computational grid to solve many large instances of these problems, allowing us to obtain highquality solutions and to verify optimality and nearoptimality of the computed solutions in various ways.
Optimization of Computer Simulation Models with Rare Events
 European Journal of Operations Research
, 1996
"... Discrete event simulation systems (DESS) are widely used in many diverse areas such as computercommunication networks, flexible manufacturing systems, project evaluation and review techniques (PERT), and flow networks. Because of their complexity, such systems are typically analyzed via Monte Ca ..."
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Cited by 53 (9 self)
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Discrete event simulation systems (DESS) are widely used in many diverse areas such as computercommunication networks, flexible manufacturing systems, project evaluation and review techniques (PERT), and flow networks. Because of their complexity, such systems are typically analyzed via Monte Carlo simulation methods. This paper deals with optimization of complex computer simulation models involving rare events. A classic example is to find an optimal (s; S) policy in a multiitem, multicommodity inventory system, when quality standards require the backlog probability to be extremely small. Our approach is based on change of the probability measure techniques, also called likelihood ratio (LR) and importance sampling (IS) methods.
The sample average approximation method applied to stochastic routing problems: a computational study
 Computational Optimization and Applications
"... Abstract. The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. ..."
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Cited by 51 (7 self)
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Abstract. The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps. We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and firststage integer variables. For each of the three problem classes, we use decomposition and branchandcut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0 % of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably nearoptimal solutions to these difficult stochastic programs using only a moderate amount of computation time. Keywords: salesman stochastic optimization, stochastic programming, stochastic routing, shortest path, traveling 1.
A SimulationBased Approach to TwoStage Stochastic Programming with Recourse
, 1996
"... In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for twosta ..."
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Cited by 36 (6 self)
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In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for twostage stochastic programming with recourse where the random data have a continuous (multivariate normal) distribution.
Stochastic Programming by Monte Carlo Simulation Methods
 Journal Statistical Planning and Inference
"... We consider in this paper stochastic programming problems which can be formulated as an optimization problem of an expected value function subject to deterministic constraints. We discuss a Monte Carlo simulation approach based on sample average approximations to a numerical solution of such problem ..."
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Cited by 23 (2 self)
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We consider in this paper stochastic programming problems which can be formulated as an optimization problem of an expected value function subject to deterministic constraints. We discuss a Monte Carlo simulation approach based on sample average approximations to a numerical solution of such problems. In particular, we give a survey of a statistical inference of the sample average estimators of the optimal value and optimal solutions of the true problem. We also discuss stopping rules and a validation analysis for such sample average approximation optimization procedures and give some illustration examples. # This work was supported, in part, by grant Grant DMI9713878 from the National Science Foundation. 1 Introduction We consider in this paper optimization problems of the form Min x#S {g(x):=IE P G(x, #)} . (1.1) Here x # IR m is a (finite dimensional) vector of decision variables, S is a closed subset of IR m representing feasible solutions of the above problem, (#, F,...
Simulationbased optimization of virtual nesting controls for network revenue management
, 2004
"... Virtual nesting is a popular capacity control strategy in network revenue management. (See Smith et al. [36].) In virtual nesting, products (itineraryfareclass combinations) are mapped ("indexed") into a relatively small number of "virtual classes" on each resource (flight leg) ..."
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Cited by 21 (2 self)
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Virtual nesting is a popular capacity control strategy in network revenue management. (See Smith et al. [36].) In virtual nesting, products (itineraryfareclass combinations) are mapped ("indexed") into a relatively small number of "virtual classes" on each resource (flight leg) of the network. Nested protection levels are then used to control the availability of these virtual classes; specifically, a product request is accepted if and only if its corresponding virtual class is available on each resource required. (See Talluri and van Ryzin [38] for a detailed discussion of virtual nesting and protection level controls.) Bertsimas and de Boer [8] recently proposed an innovative simulationbased optimization method for computing protection levels in a virtual nesting control scheme. In contrast to traditional heuristic methods, their approach more accurately approximates the true network revenues generated by the virtual nesting controls. However, because it is based on a discrete model of capacity and demand, the method has both computational and theoretical limitations. In particular, it uses firstdifference estimates, which are computationally complex to calculate exactly. These gradient estimates are then used in a steepest ascent type algorithm, which, for discrete problems, has no guarantee of convergence.
On choosing parameters in retrospectiveapproximation algorithms for simulationoptimization
 Proceedings of the 2006 Winter Simulation Conference. Institute of Electrical and Electronics Engineers: Piscataway
"... The Stochastic RootFinding Problem is that of finding a zero of a vectorvalued function known only through a stochastic simulation. The SimulationOptimization Problem is that of locating a realvalued function’s minimum, again with only a stochastic simulation that generates function estimates. ..."
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Cited by 13 (6 self)
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The Stochastic RootFinding Problem is that of finding a zero of a vectorvalued function known only through a stochastic simulation. The SimulationOptimization Problem is that of locating a realvalued function’s minimum, again with only a stochastic simulation that generates function estimates. Retrospective Approximation (RA) is a samplepath technique for solving such problems, where the solution to the underlying problem is approached via solutions to a sequence of approximate deterministic problems, each of which is generated using a specified sample size, and solved to a specified error tolerance. Our primary focus, in this paper, is providing guidance on choosing the sequence of sample sizes and error tolerances in RA algorithms. We first present an overview of the conditions that guarantee the correct convergence of RA’s iterates. Then we characterize a class of errortolerance and samplesize sequences that are superior to others in a certain precisely defined sense. We also identify and recommend members of this class, and provide a numerical example illustrating the key results. 1
On rates of convergence for stochastic optimization problems under nonI.I.D. sampling
, 2006
"... In this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. This is a well studied problem in case the samples are ..."
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Cited by 12 (1 self)
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In this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. This is a well studied problem in case the samples are independent and identically distributed (i.e., when standard Monte Carlo is used); here, we study the case where that assumption is dropped. Broadly speaking, our results show that, under appropriate assumptions, the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case, in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation. Our motivation for the study arises from two types of sampling methods that have been widely used in the Statistics literature. One is Latin Hypercube Sampling (LHS), a stratified sampling method originally proposed in the seventies by McKay, Beckman, and Conover (1979). The other is the class of quasiMonte Carlo (QMC) methods, which have become popular especially after the work of Niederreiter (1992). The advantage of such methods is that they typically yield pointwise estimators which not only have lower variance than standard Monte Carlo but also possess better rates of convergence. Thus, it is important to study the use of these techniques in samplingbased optimization. The novelty of our work arises from the fact that, while there has been some work on the use of variance reduction techniques and QMC methods in stochastic optimization, none of the existing work — to the best of our knowledge — has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem. We present numerical results for some twostage stochastic programs from the literature to illustrate the discussed ideas.
A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing
 IEEE Transactions on Circuits and SystemsI
, 2004
"... A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is th ..."
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Cited by 12 (4 self)
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A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is the minimum time required to complete all activities. In a stochastic activity network (SAN), the durations of the activities and the makespan are random variables. The analysis of SANs is quite involved, but can be carried out numerically by Monte Carlo analysis. This paper concerns the optimization of a SAN, i.e., the choice of some design variables that affect the probability distributions of the activity durations. We concentrate on the problem of minimizing a quantile (e.g., 95%) of the makespan, subject to constraints on the variables. This problem has many applications, ranging from project management to digital integrated circuit (IC) sizing (the latter being our motivation). While there are effective methods for optimizing DANs, the SAN optimization problem is much more difficult; the few existing methods cannot handle largescale problems.