On Complexity of Stochastic Programming Problems

Shapiro, Alexander; Nemirovski, Arkadi

doi:10.1007/0-387-26771-9_4

Cited by 246 publications

(212 citation statements)

References 51 publications

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“…Whereas a stochastic optimization approach addresses the issue of uncertain parameters, it is by and large computationally intractable. Shapiro and Nemirovski [21] give hardness results for two-stage and multistage stochastic optimization problems where they show that multistage stochastic optimization is computationally intractable even if approximate solutions are desired. Furthermore, to solve a two-stage stochastic optimization problem, Shapiro and Nemirovski [21] present an approximate sampling based algorithm where a sufficiently large number of scenarios (depending on the variance of the objective function and the desired accuracy level) are sampled from the assumed distribution and the solution to the resulting sampled problem is argued to provide an approximate solution to the original problem.…”

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confidence: 99%

On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

Bertsimas

Goyal

2010

Mathematics of OR

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We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right-hand side of the constraints is uncertain and belongs to a symmetric uncertainty set (such as hypercube, ellipsoid or norm ball) and the probability measure is also symmetric, then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two-stage stochastic problem. Furthermore, we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric. We refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two-stage stochastic problem as the stochasticity gap. We also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive.If both the objective coefficients and right-hand side are uncertain, we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric. However, we prove that the adaptability gap (ratio of optimal cost of the robust problem and the optimal cost of a two-stage fully adaptable problem) is at most four even if both the objective coefficients and the right-hand side of the constraints are uncertain and belong to a symmetric uncertainty set. The bound holds for the class of positive uncertainty sets as well. Moreover, if the uncertainty set is a hypercube (special case of a symmetric set), the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain. 1. Introduction. In most real-world problems, several parameters are uncertain at the optimization phase and a solution obtained through a deterministic optimization approach might be sensitive to even slight perturbations in the problem parameters, possibly rendering it highly suboptimal or infeasible. Stochastic optimization that was introduced as early as Dantzig [10] has been extensively studied in the literature to address uncertainty. A stochastic optimization approach assumes a probability distribution over the uncertain parameters and tries to compute a (two-stage or a multistage) solution that optimizes the expected value of the objective function. We refer the reader to several textbooks including Infanger [15] give hardness results for two-stage and multistage stochastic optimization problems where they show that multistage stochastic optimization is computationally intractable even if approximate solutions are desired. Furthermore, to solve a two-stage stochastic optimization problem, Shapiro and Nemirovski [21] present an approximate sampling based algorithm where a sufficiently large number of scenarios (depending on the variance of the objective function and the desi...

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confidence: 99%

On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

Bertsimas

Goyal

2010

Mathematics of OR

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“…Let (x, r) be a feasible solution to the LP relaxation of (21). Observe that for each element e ∈ U , we either have S:e∈S x S ≥ 1/2 or S:e∈S r A,S ≥ 1/2 for every scenario A that contains e. Now, define:…”

Section: Covering Problemsmentioning

confidence: 99%

“…Under some mild assumptions, it has been shown [14,21] that the optimal value of (2) is a good approximation to that of (1) with high probability, and that the number of samples N can be bounded. Unfortunately, the bound on N depends on the maximum variance V (over all x ∈ X) of the random variables q(x, ω), which need not be polynomially bounded by the input size.…”

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confidence: 99%

Stochastic Combinatorial Optimization with Controllable Risk Aversion Level

Zhang

2009

Mathematics of OR

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Most of the recent work on 2-stage stochastic combinatorial optimization problems have focused on the minimization of the expected cost or the worst-case cost of the solution. Those two objectives can be viewed as two extreme ways of handling risk. In this paper we propose to use an one-parameter family of functionals to interpolate between them. Although such a family has been used in the mathematical finance and stochastic programming literature before, its use in the context of approximation algorithms seems new. We show that under standard assumptions, a broad class of risk-adjusted 2-stage stochastic programs can be efficiently treated by the Sample Average Approximation (SAA) method. In particular, our result shows that it is computationally feasible to incorporate some degree of robustness even when the underlying distribution can only be accessed in a black-box fashion. We also show that when combined with suitable rounding procedures, our result yields new approximation algorithms for many risk-adjusted 2-stage stochastic combinatorial optimization problems under the black-box setting.

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“…As was pointed out in [43], solving and analyzing the SAA counterparts for multistage stochastic models seem to be very hard in general. Instead of solving the SAA counterpart of the multiperiod model, we propose a dynamic programming framework that departs from previous sampling-based algorithms.…”

Section: Introductionmentioning

confidence: 99%

“…Kleywegt, Shapiro and Homem-De-Mello [24], Shapiro and Nemirovski [43] and Shapiro [40] have considered the SAA in a general setting of two-stage discrete stochastic optimization models (see [35] for discussion on two-stage stochastic models). They have shown that the optimal value of the SAA problem converges to the optimal value of the original problem with probability 1 as the number of samples grows to infinity.…”

Section: Introductionmentioning

confidence: 99%

Provably Near-Optimal Sampling-Based Policies for Stochastic Inventory Control Models

2007

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In this paper, we consider two fundamental inventory models, the single-period newsvendor problem and its multi-period extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. Under the assumption that the demand distributions are given explicitly, these models are well-studied and relatively straightforward to solve. However, in most real-life scenarios, the true demand distributions are not available or they are too complex to work with. Thus, a sampling-driven algorithmic framework is very attractive, both in practice and in theory.We shall describe how to compute sampling-based policies, that is, policies that are computed based only on observed samples of the demands without any access to, or assumptions on, the true demand distributions. Moreover, we establish bounds on the number of samples required to guarantee that with high probability, the expected cost of the sampling-based policies is arbitrarily close (i.e., with arbitrarily small relative error) compared to the expected cost of the optimal policies which have full access to the demand distributions. The bounds that we develop are general, easy to compute and do not depend at all on the specific demand distributions.Key words: Inventory, Approximation ; Sampling ;Algorithms ; Nonparametric MSC2000 Subject Classification: Primary: 90B05 , ; Secondary: 62G99 , OR/MS subject classification: Primary: inventory/production , approximation/heuristics ; Secondary: production/scheduling , approximation/heuristics, learning 1. Introduction In this paper, we address two fundamental models in stochastic inventory theory, the single-period newsvendor model and its multiperiod extension, under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. Under the assumption that the demand distributions are specified explicitly, these models are well-studied and usually straightforward to solve. However, in most real-life scenarios, the true demand distributions are not available or they are too complex to work with. Usually, the information that is available comes from historical data, from a simulation model, and from forecasting and market analysis of future trends in the demands. Thus, we believe that a sampling-driven algorithmic framework is very attractive, both in practice and in theory. In this paper, we shall describe how to compute sampling-based policies, that is, policies that are computed based only on observed samples of the demands without any access to and assumptions on the true demand distributions. This is usually called a non-parametric approach. Moreover, we shall prove that the quality (expected cost) of these policies is very close to that of the optimal policies that are defined with respect to the true underlying demand distributions.

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On Complexity of Stochastic Programming Problems

Cited by 246 publications

References 51 publications

On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

On the Power of Robust Solutions in Two-Stage Stochastic and Adaptive Optimization Problems

Stochastic Combinatorial Optimization with Controllable Risk Aversion Level

Provably Near-Optimal Sampling-Based Policies for Stochastic Inventory Control Models

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