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Adjustable robust optimization through multi-parametric programming

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Abstract

Adjustable robust optimization (ARO) involves recourse decisions (i.e. reactive actions after the realization of the uncertainty, ‘wait-and-see’) as functions of the uncertainty, typically posed in a two-stage stochastic setting. Solving the general ARO problems is challenging, therefore ways to reduce the computational effort have been proposed, with the most popular being the affine decision rules, where ‘wait-and-see’ decisions are approximated as affine adjustments of the uncertainty. In this work we propose a novel method for the derivation of generalized affine decision rules for linear mixed-integer ARO problems through multi-parametric programming, that lead to the exact and global solution of the ARO problem. The problem is treated as a multi-level programming problem and it is then solved using a novel algorithm for the exact and global solution of multi-level mixed-integer linear programming problems. The main idea behind the proposed approach is to solve the lower optimization level of the ARO problem parametrically, by considering ‘here-and-now’ variables and uncertainties as parameters. This will result in a set of affine decision rules for the ‘wait-and-see’ variables as a function of ‘here-and-now’ variables and uncertainties for their entire feasible space. A set of illustrative numerical examples are provided to demonstrate the potential of the proposed novel approach.

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Acknowledgements

We are grateful to Prof. Fengqi You and his student Chao Ning for providing us with the solutions of the numerical examples using the C&C approach. We are also grateful to NSF Projects PAROC (Award No.1705423) and INFEWS (Award No. 1739977), Texas A&M University and Texas A&M Energy Institute for financial support.

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  1. Texas A&M Energy Institute, Texas A&M University, College Station, TX, USA

    Styliani Avraamidou & Efstratios N. Pistikopoulos

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  1. Styliani Avraamidou
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  2. Efstratios N. Pistikopoulos
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Correspondence to Efstratios N. Pistikopoulos.

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Avraamidou, S., Pistikopoulos, E.N. Adjustable robust optimization through multi-parametric programming. Optim Lett 14, 873–887 (2020). https://doi.org/10.1007/s11590-019-01438-5

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