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General Robust-Optimization Formulation for Nonlinear Programming

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  • Y. Zhang

    (Rice University)

Abstract

Most research in robust optimization has been focused so far on inequality-only, convex conic programming with simple linear models for the uncertain parameters. Many practical optimization problems, however, are nonlinear and nonconvex. Even in linear programming, the coefficients may still be nonlinear functions of the uncertain parameters. In this paper, we propose robust formulations that extend the robust-optimization approach to a general nonlinear programming setting with parameter uncertainty involving both equality and inequality constraints. The proposed robust formulations are valid in a neighborhood of a given nominal parameter value and are robust to the first-order, thus suitable for applications where reasonable parameter estimations are available and uncertain variations are moderate.

Suggested Citation

  • Y. Zhang, 2007. "General Robust-Optimization Formulation for Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 132(1), pages 111-124, January.
    • Handle: RePEc:spr:joptap:v:132:y:2007:i:1:d:10.1007_s10957-006-9082-z
    DOI: 10.1007/s10957-006-9082-z
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    Cited by:

    1. Gabrel, Virginie & Murat, Cécile & Thiele, Aurélie, 2014. "Recent advances in robust optimization: An overview," European Journal of Operational Research, Elsevier, vol. 235(3), pages 471-483.
    2. Alabi, Tobi Michael & Lu, Lin & Yang, Zaiyue, 2021. "Stochastic optimal planning scheme of a zero-carbon multi-energy system (ZC-MES) considering the uncertainties of individual energy demand and renewable resources: An integrated chance-constrained and," Energy, Elsevier, vol. 232(C).
    3. C. Cromvik & M. Patriksson, 2010. "On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 1: Theory," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 461-478, March.
    4. Emilio Carrizosa & Frédéric Messine, 2021. "An interval branch and bound method for global Robust optimization," Journal of Global Optimization, Springer, vol. 80(3), pages 507-522, July.
    5. E. T. Hale & Y. Zhang, 2007. "Case Studies for a First-Order Robust Nonlinear Programming Formulation," Journal of Optimization Theory and Applications, Springer, vol. 134(1), pages 27-45, July.
    6. Hsien-Chung Wu, 2019. "Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems," Mathematics, MDPI, vol. 7(5), pages 1-50, May.
    7. Hsien-Chung Wu, 2021. "Robust Solutions for Uncertain Continuous-Time Linear Programming Problems with Time-Dependent Matrices," Mathematics, MDPI, vol. 9(8), pages 1-52, April.
    8. Soleimanian, Azam & Salmani Jajaei, Ghasemali, 2013. "Robust nonlinear optimization with conic representable uncertainty set," European Journal of Operational Research, Elsevier, vol. 228(2), pages 337-344.
    9. Theodor Komann & Michael Wiesheu & Stefan Ulbrich & Sebastian Schöps, 2024. "Robust Design Optimization of Electric Machines with Isogeometric Analysis," Mathematics, MDPI, vol. 12(9), pages 1-18, April.
    10. Dimitris Bertsimas & Omid Nohadani & Kwong Meng Teo, 2010. "Robust Optimization for Unconstrained Simulation-Based Problems," Operations Research, INFORMS, vol. 58(1), pages 161-178, February.

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