Linearized Robust Counterparts of Two-Stage Robust Optimization Problems with Applications in Operations Management

Ardestani-Jaafari, Amir; Delage, Erick

doi:10.1287/ijoc.2020.0959

Cited by 13 publications

(8 citation statements)

References 27 publications

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“…We circumvent this exponential growth by utilizing tractable conservative approximations inspired by lift-and-project convexification techniques in global optimization [33,35,47]. Closest in spirit to our work are the papers of [2,27,56] who consider the case where the second-stage loss Q(x, ξ) is the optimal value of a linear program with uncertain right-hand sides and the support set Ξ is a polytope. In this setting, [27,56] reformulate (2) as a copositive cone program and then approximate this using semidefinite programming, whereas [2] provide approximations by leveraging reformulation-linearization techniques from bilinear programming.…”

Section: Contributionsmentioning

confidence: 99%

“…Closest in spirit to our work are the papers of [2,27,56] who consider the case where the second-stage loss Q(x, ξ) is the optimal value of a linear program with uncertain right-hand sides and the support set Ξ is a polytope. In this setting, [27,56] reformulate (2) as a copositive cone program and then approximate this using semidefinite programming, whereas [2] provide approximations by leveraging reformulation-linearization techniques from bilinear programming. In this paper, we state a result that enables the encapsulation of these seemingly disparate methods in a common framework.…”

Section: Contributionsmentioning

confidence: 99%

“…These hierarchies were originally proposed for (pure or mixed-) integer linear sets and later extended to mixed-integer convex sets in [18,48]. Our proposal is to use an intermediate relaxation Z t of any such hierarchy to outer approximate cl conv(Z), which results in an outer approximation of the convex hull reformulation (7) and, hence, of the distributionally robust two-stage problem (2).…”

Section: Lift-and-project Approximationsmentioning

confidence: 99%

“…Recall from Section 5.1 that higher values of ψ increase the probability of line failures, whereas higher values of φ increase the penalty cost for failing to satisfy power demand due to transmission line failures. Figure 7 shows the relative improvement of the distributionally robust two-stage model (2) over the sample average approximation for various choices of ψ and φ. For brevity, we report results only for the 30-bus instance; the high-level insights do not change for other instances.…”

Section: Sensitivity Analysismentioning

confidence: 99%

“…This allows us to exploit existing tools on characterizing the convex hulls of MICP sets. Section A.1 highlights several problem instances (i.e., sufficient conditions) where this convex hull is efficiently computable and hence the distributionally robust two-stage problem (2) is tractable. In general, however, the presence of discrete random parameters makes the problem intractable even when the second-stage problem Q(x, ξ) is benign.…”

Section: Appendix a Complexity Analysismentioning

confidence: 99%

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Data-driven two-stage conic optimization with zero-one uncertainties

Subramanyam¹,

Tonbari²,

Kim³

2020

Preprint

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We address high-dimensional zero-one random parameters in two-stage convex conic optimization problems. Such parameters typically represent failures of network elements and constitute rare, high-impact random events in several applications. Given a sparse training dataset of the parameters, we motivate and study a distributionally robust formulation of the problem using a Wasserstein ambiguity set centered at the empirical distribution. We present a simple, tractable, and conservative approximation of this problem that can be efficiently computed and iteratively improved. Our method relies on a reformulation that optimizes over the convex hull of a mixed-integer conic programming representable set, followed by an approximation of this convex hull using lift-and-project techniques. We illustrate the practical viability and strong out-of-sample performance of our method on nonlinear optimal power flow problems affected by random contingencies, and we report improvements of up to 20% over existing methods. MotivationThis work is motivated by optimization problems arising in applications that are affected by an extremely large, yet finite, number of rare, high-impact random events. In particular, we are motivated by applications in which the decision-relevant random events consist of high-dimensional binary outcomes. Such applications are ubiquitous in network optimization, where the uncertain 1

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Section: Contributionsmentioning

confidence: 99%

Section: Contributionsmentioning

confidence: 99%

Section: Lift-and-project Approximationsmentioning

confidence: 99%

Section: Sensitivity Analysismentioning

confidence: 99%

Section: Appendix a Complexity Analysismentioning

confidence: 99%

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Data-driven two-stage conic optimization with zero-one uncertainties

Subramanyam¹,

Tonbari²,

Kim³

2020

Preprint

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Adjustability in robust linear optimization

Wei,

Zhang

2024

Math. Program.

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We investigate the concept of adjustability—the difference in objective values between two types of dynamic robust optimization formulations: one where (static) decisions are made before uncertainty realization, and one where uncertainty is resolved before (adjustable) decisions. This difference reflects the value of information and decision timing in optimization under uncertainty, and is related to several other concepts such as the optimality of decision rules in robust optimization. We develop a theoretical framework to quantify adjustability based on the input data of a robust optimization problem with a linear objective, linear constraints, and fixed recourse. We make very few additional assumptions. In particular, we do not assume constraint-wise separability or parameter nonnegativity that are commonly imposed in the literature for the study of adjustability. This allows us to study important but previously under-investigated problems, such as formulations with equality constraints and problems with both upper and lower bound constraints. Based on the discovery of an interesting connection between the reformulations of the static and fully adjustable problems, our analysis gives a necessary and sufficient condition—in the form of a theorem-of-the-alternatives—for adjustability to be zero when the uncertainty set is polyhedral. Based on this sharp characterization, we provide two efficient mixed-integer optimization formulations to verify zero adjustability. Then, we develop a constructive approach to quantify adjustability when the uncertainty set is general, which results in an efficient and tight poly-time algorithm to bound adjustability. We demonstrate the efficiency and tightness via both theoretical and numerical analyses.

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A Lagrangian dual method for two-stage robust optimization with binary uncertainties

Subramanyam

2022

Optim Eng

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We provide a correction to the sufficient conditions under which closed-form expressions for the optimal Lagrange multiplier are provided in [4]. We first present a simple counterexample where the original conditions are insufficient, highlight where the original proof fails, and then provide modified conditions along with a correct proof of their validity. Finally, although the original paper discusses modifications to their method for problems that may not satisfy any sufficient conditions, we substantiate that discussion along two directions. We first show that computing an optimal Lagrange multiplier can still be done in polynomial time. We then provide complete and correct versions of the corresponding Benders and column-and-constraint generation algorithms in which the original method is used. We also discuss the implications of our findings on computational performance.

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Linearized Robust Counterparts of Two-Stage Robust Optimization Problems with Applications in Operations Management

Cited by 13 publications

References 27 publications

Data-driven two-stage conic optimization with zero-one uncertainties

Data-driven two-stage conic optimization with zero-one uncertainties

Adjustability in robust linear optimization

A Lagrangian dual method for two-stage robust optimization with binary uncertainties

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