Lectures on Stochastic Programming

Shapiro, Alexander; Dentcheva, Darinka; Ruszczyński, Andrzej

doi:10.1137/1.9780898718751

Cited by 1,400 publications

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“…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], Kall and Wallace [16], Prékopa [17], Birge and Louveaux [9], Shapiro [20], Shapiro et al [22] and the references therein for a comprehensive view of stochastic optimization. Whereas a stochastic optimization approach addresses the issue of uncertain parameters, it is by and large computationally intractable.…”

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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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“…Our notation and framework are closely in line with that of [Shapiro et al, 2009], to which we direct the reader for more details.…”

Section: Probabilistic Modelmentioning

confidence: 99%

“…From these relations, we have that µ C pY q " max qPQ q T Y , wherẽ Q has the following product form structure [Shapiro et al, 2009]:…”

Section: Technical Proofsmentioning

confidence: 99%

Tight Approximations of Dynamic Risk Measures

Iancu

Petrik²,

Subramanian³

2015

Mathematics of OR

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This paper compares two different frameworks recently introduced in the literature for measuring risk in a multi-period setting. The first corresponds to applying a single coherent risk measure to the cumulative future costs, while the second involves applying a composition of one-step coherent risk mappings. We summarize the relative strengths of the two methods, characterize several necessary and sufficient conditions under which one of the measurements always dominates the other, and introduce a metric to quantify how close the two risk measures are.Using this notion, we address the question of how tightly a given coherent measure can be approximated by lower or upper-bounding compositional measures. We exhibit an interesting asymmetry between the two cases: the tightest possible upper-bound can be exactly characterized, and corresponds to a popular construction in the literature, while the tightest-possible lower bound is not readily available. We show that testing domination and computing the approximation factors is generally NP-hard, even when the risk measures in question are comonotonic and law-invariant. However, we characterize conditions and discuss several examples where polynomial-time algorithms are possible. One such case is the well-known Conditional Value-at-Risk measure, which is further explored in our companion paper Huang et al. [2012]. Our theoretical and algorithmic constructions exploit interesting connections between the study of risk measures and the theory of submodularity and combinatorial optimization, which may be of independent interest.

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“…For any x ∈ Vk we have (for the same random sample) that the empirical cdf of G(x, ·) dominates the empirical cdf of Gk(·), and hence (cf., [8,Theorem 6.28])…”

Section: Convergence Of Statistical Estimates Of Risk Averse Stochastmentioning

confidence: 99%

“…In the context of solving the optimization problem (1.4) this is the approach of the so-called Sample Average Approximation (SAA) method (see, e.g., [8,ter 5]). Although conceptually different these two applications involve the same statistical inference.…”

Section: Introductionmentioning

confidence: 99%

Consistency of Sample Estimates of Risk Averse Stochastic Programs

Shapiro¹

2013

J. Appl. Probab.

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Abstract. In this paper we study asymptotic consistency of law invariant convex risk measures and the corresponding risk averse stochastic programming problems for independent identically distributed data. Under mild regularity conditions we prove a Law of Large Numbers and epiconvergence of the corresponding statistical estimators. This can be applied in a straightforward way to establishing convergence w.p.1 of sample based estimators of risk averse stochastic programming problems.

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Lectures on Stochastic Programming

Cited by 1,400 publications

References 0 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

Tight Approximations of Dynamic Risk Measures

Consistency of Sample Estimates of Risk Averse Stochastic Programs

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