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A Framework for Solving Chance-Constrained Linear Matrix Inequality Programs

Author

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  • Roya Karimi

    (Department of Systems and Industrial Engineering, University of Arizona, Tucson, Arizona 85721)

  • Jianqiang Cheng

    (Department of Systems and Industrial Engineering, University of Arizona, Tucson, Arizona 85721)

  • Miguel A. Lejeune

    (Department of Decisions Sciences, George Washington Business School, George Washington University, Washington, District of Columbia 20052; Department of Electrical and Computer Engineering, School of Engineering and Applied Science, George Washington University, Washington, District of Columbia 20052)

Abstract

We propose a novel partial sample average approximation (PSAA) framework to solve the two main types of chance-constrained linear matrix inequality (CCLMI) problems: CCLMI with random technology matrix and CCLMI with random right-hand side. We propose a series of computationally tractable PSAA-based approximations for CCLMI problems, analyze their properties, and derive sufficient conditions that ensure convexity for the two most popular—normal and uniform—continuous distributions. We derive several semidefinite programming PSAA reformulations efficiently solved by off-the-shelf solvers and design a sequential convex approximation method for the PSAA formulations containing bilinear matrix inequalities. The proposed methods can be generalized to other continuous random variables whose cumulative distribution function can be easily computed. We carry out a comprehensive numerical study on three practical CCLMI problems: robust truss topology design, calibration, and robust control. The tests attest to the superiority of the PSAA reformulation and algorithmic framework over the scenario and sample average approximation methods. Summary of Contribution: In line with the mission and scope of IJOC, we study an important type of optimization problems, chance-constrained linear matrix inequality (CCLMI) problems, which require stochastic linear matrix inequality (LMI) constraints to be satisfied with high probability. To solve CCLMI problems, we propose a novel partial sample average approximation (PSAA) framework: (i) develop a series of computationally tractable PSAA-based approximations for CCLMI problems, (ii) analyze their properties, (iii) derive sufficient conditions ensuring convexity, and (iv) design a sequential convex approximation method. We evaluate our proposed method via a comprehensive numerical study on three practical CCLMI problems. The tests attest the superiority of the PSAA reformulation and algorithmic framework over standard benchmarks.

Suggested Citation

  • Roya Karimi & Jianqiang Cheng & Miguel A. Lejeune, 2021. "A Framework for Solving Chance-Constrained Linear Matrix Inequality Programs," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1015-1036, July.
    • Handle: RePEc:inm:orijoc:v:33:y:2021:i:3:p:1015-1036
    DOI: 10.1287/ijoc.2020.0982
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    References listed on IDEAS

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    1. Wenqing Chen & Melvyn Sim & Jie Sun & Chung-Piaw Teo, 2010. "From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization," Operations Research, INFORMS, vol. 58(2), pages 470-485, April.
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    4. Jianqiang Cheng & Richard Li-Yang Chen & Habib N. Najm & Ali Pinar & Cosmin Safta & Jean-Paul Watson, 2018. "Chance-constrained economic dispatch with renewable energy and storage," Computational Optimization and Applications, Springer, vol. 70(2), pages 479-502, June.
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