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Complexity Results and Effective Algorithms for Worst-Case Linear Optimization Under Uncertainties

Author

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  • Hezhi Luo

    (Department of Mathematics, College of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310032, China)

  • Xiaodong Ding

    (Department of Applied Mathematics, College of Science, Zhejiang University of Technology, Hangzhou, Zhejiang 310032, China)

  • Jiming Peng

    (Department of Industrial Engineering, University of Houston, Houston, Texas 77204)

  • Rujun Jiang

    (School of Data Science, Fudan University, Shanghai 200433, China)

  • Duan Li

    (School of Data Science, City University of Hong Kong, Hong Kong)

Abstract

In this paper, we consider the so-called worst-case linear optimization (WCLO) with uncertainties on the right-hand side of the constraints. Such a problem often arises in applications such as in systemic risk estimation in finance and stochastic optimization. We first show that the WCLO problem with the uncertainty set corresponding to the l p -norm ((WCLO p )) is NP-hard for p ɛ (1,∞). Second, we combine several simple optimization techniques, such as the successive convex optimization method, quadratic convex relaxation, initialization, and branch-and-bound (B&B), to develop an algorithm for (WCLO 2 ) that can find a globally optimal solution to (WCLO 2 ) within a prespecified ε-tolerance. We establish the global convergence of the algorithm and estimate its complexity. We also develop a finite B&B algorithm for (WCLO ∞ ) to identify a global optimal solution to the underlying problem, and establish the finite convergence of the algorithm. Numerical experiments are reported to illustrate the effectiveness of our proposed algorithms in finding globally optimal solutions to medium and large-scale WCLO instances.

Suggested Citation

  • Hezhi Luo & Xiaodong Ding & Jiming Peng & Rujun Jiang & Duan Li, 2021. "Complexity Results and Effective Algorithms for Worst-Case Linear Optimization Under Uncertainties," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 180-197, January.
    • Handle: RePEc:inm:orijoc:v:33:y:2021:i:1:p:180-197
    DOI: 10.1287/ijoc.2019.0941
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    References listed on IDEAS

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    Cited by:

    1. Xiaodong Ding & Hezhi Luo & Huixian Wu & Jianzhen Liu, 2021. "An efficient global algorithm for worst-case linear optimization under uncertainties based on nonlinear semidefinite relaxation," Computational Optimization and Applications, Springer, vol. 80(1), pages 89-120, September.

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