Saddle point approximation approaches for two-stage robust optimization problems

Abstract

This paper aims to present improvable and computable approximation approaches for solving the two-stage robust optimization problem, which arises from various applications such as optimal energy management and production planning. Based on sampling finite number scenarios of uncertainty, we can obtain a lower bound approximation and show that the corresponding solution is at least \({\varepsilon }\)-level feasible. Moreover, piecewise linear decision rules (PLDRs) are also introduced to improve the upper bound that obtained by the widely-used linear decision rule. Furthermore, we show that both the lower bound and upper bound approximation problems can be reformulated into solvable saddle point problems and consequently be solved by the mirror descent method.

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Acknowledgements

We would like to thank Professor Xu Huan for his constructive suggestions on the idea of solving the two-stage robust optimization by using the randomized approach.

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Correspondence to Chang Fang.

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This author is supported by the National Natural Science Foundation of China (No. 71801005), the Natural Science Foundation of Anhui Province (Nos. 1808085QG227, 1608085MG152), and the Social Sciences Foundation of Anhui Province (No. AHSKQ2016D28)

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Zhang, N., Fang, C. Saddle point approximation approaches for two-stage robust optimization problems. J Glob Optim 78, 651–670 (2020). https://doi.org/10.1007/s10898-019-00836-4

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Keywords

  • Two-stage robust optimization
  • Randomized approach
  • Piecewise linear decision rule
  • Saddle point problem
  • Mirror descent algorithm