When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?

A. Marandi, D. den Hertog

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Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.


Keywords
Robust optimization Adjustable robust optimization Constraint-wise uncertainty Hybrid uncertainty
TaalEngels
Pagina's555-568
TijdschriftMathematical Programming
Volume170
Nummer van het tijdschrift2
DOI's
StatusGepubliceerd - 1 aug 2018

Vingerafdruk

Robust Optimization
Optimization Problem
Uncertainty
Uncertain Parameters
Decision Rules
Compact Set
Convex Sets

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    title = "When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?",
    abstract = "Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.KeywordsRobust optimization Adjustable robust optimization Constraint-wise uncertainty Hybrid uncertainty",
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    author = "A. Marandi and {den Hertog}, D.",
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    When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent? / Marandi, A.; den Hertog, D.

    In: Mathematical Programming, Vol. 170, Nr. 2, 01.08.2018, blz. 555-568.

    Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

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    AU - den Hertog,D.

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    N2 - Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.KeywordsRobust optimization Adjustable robust optimization Constraint-wise uncertainty Hybrid uncertainty

    AB - Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable variables and concave with respect to the uncertain parameters, the adjustable variables lie in a convex and compact set and the uncertainty set is convex and compact, then robust solutions are also optimal for the corresponding ARO problem. Furthermore, we prove that if some of the uncertain parameters are constraint-wise and the rest are not, then under a similar set of assumptions there is an optimal decision rule for the ARO problem that does not depend on the constraint-wise uncertain parameters. Also, we show for a class of problems that using affine decision rules that depend on all of the uncertain parameters yields the same optimal objective value as when the rules depend solely on the non-constraint-wise uncertain parameters. Finally, we illustrate the usefulness of these results by applying them to convex quadratic and conic quadratic problems.KeywordsRobust optimization Adjustable robust optimization Constraint-wise uncertainty Hybrid uncertainty

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