Optimality of Affine Policies in Multistage Robust Optimization

Bertsimas, Dimitris; Iancu, Dan A.; Parrilo, Pablo A.

doi:10.1287/moor.1100.0444

Cited by 191 publications

(111 citation statements)

References 25 publications

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“…This approach was first considered in Rockafellar and Wets [19] in the context of stochastic optimization, and then in robust optimization (Ben-Tal et al [2]), and extended to linear systems theory (Ben-Tal et al [1]). In a recent paper, Bertsimas et al [8] consider a one-dimensional, box-constrained multistage robust optimization problem and show that an affine policy is optimal in this setting. However, in general an affine policy does not necessarily provide a good approximation to the adaptive problem (Bertsimas and Goyal [4]).…”

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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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“…Most applications of ARO restrict decision rules to affine functions, which is referred to as affinely adjustable robust optimization (AARO) [2]. Affine decision rules are known to perform optimal or nearly optimal in many situations [1,4]. However, once again, this observation is with respect to the worst-case objective value, and not for the mean objective value.…”

Section: Toy Examplesmentioning

confidence: 99%

The impact of the existence of multiple adjustable robust solutions

2016

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In this note we show that multiple solutions exist for the productioninventory example in the seminal paper on adjustable robust optimization in Ben-Tal et al. (Math Program 99(2):351-376, 2004). All these optimal robust solutions have the same worst-case objective value, but the mean objective values differ up to 21.9 % and for individual realizations this difference can be up to 59.4 %. We show via additional experiments that these differences in performance become negligible when using a folding horizon approach. The aim of this paper is to convince users of adjustable robust optimization to check for existence of multiple solutions. Using the production-inventory example and an illustrative toy example we deduce three important implications of the existence of multiple optimal robust solutions. First, if one neglects this existence of multiple solutions, then one can wrongly conclude that the adjustable robust solution does not outperform the nonadjustable robust solution. Second, even when it is a priori known that the adjustable and nonadjustable robust solutions are equivalent on worst-case objective value, they might still differ on the mean objective value. Third, even if it is known that affine decision rules yield (near) optimal performance in the adjustable robust optimization setting, then still nonlinear decision rules can yield much better mean objective values.This

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“…Ben-Tal et al (2005) demonstrate that an affinely adjustable robust counterpart can be remarkably effective in minimizing the worst-case objective of a multiperiod inventory control problem. Bertsimas et al (2010) show that the affinely adjustable robust counterpart can be optimal in some situations. See and Sim (2010) demonstrate the effectiveness of piecewise linear decision rules in minimizing the expected objective of a multiperiod inventory control problem under stochastic demand with correlation.…”

Section: Introductionmentioning

confidence: 98%

Robust Storage Assignment in Unit-Load Warehouses

2012

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A ssigning products to and retrieving them from proper storage locations are crucial decisions in minimizing the operating cost of a unit-load warehouse. The problem becomes intractable when the warehouse faces variable supply and uncertain demand in a multiperiod setting. We assume a factor-based demand model in which demand for each product in each period is affinely dependent on some uncertain factors. The distributions of these factors are only partially characterized. We introduce a robust optimization model that minimizes the worst-case expected total travel in the warehouse with distributional ambiguity of demand. Under a linear decision rule, we obtain a storage and retrieval policy by solving a moderate-size linear optimization problem. Surprisingly, despite imprecise specification of demand distributions, our computational studies suggest that the linear policy achieves close to the expected value given perfect information and significantly outperforms existing heuristics in the literature.

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Optimality of Affine Policies in Multistage Robust Optimization

Cited by 191 publications

References 25 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

The impact of the existence of multiple adjustable robust solutions

Robust Storage Assignment in Unit-Load Warehouses

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