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Calculating Radius of Robust Feasibility of Uncertain Linear Conic Programs via Semi-definite Programs

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  • M. A. Goberna

    (University of Alicante)

  • V. Jeyakumar

    (University of New South Wales)

  • G. Li

    (University of New South Wales)

Abstract

The radius of robust feasibility provides a numerical value for the largest possible uncertainty set that guarantees robust feasibility of an uncertain linear conic program. This determines when the robust feasible set is non-empty. Otherwise, the robust counterpart of an uncertain program is not well defined as a robust optimization problem. In this paper, we address a key fundamental question of robust optimization: How to compute the radius of robust feasibility of uncertain linear conic programs, including linear programs? We first provide computable lower and upper bounds for the radius of robust feasibility for general uncertain linear conic programs under the commonly used ball uncertainty set. We then provide important classes of linear conic programs where the bounds are calculated by finding the optimal values of related semi-definite linear programs, among them uncertain semi-definite programs, uncertain second-order cone programs and uncertain support vector machine problems. In the case of an uncertain linear program, the exact formula allows us to calculate the radius by finding the optimal value of an associated second-order cone program.

Suggested Citation

  • M. A. Goberna & V. Jeyakumar & G. Li, 2021. "Calculating Radius of Robust Feasibility of Uncertain Linear Conic Programs via Semi-definite Programs," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 597-622, May.
  • Handle: RePEc:spr:joptap:v:189:y:2021:i:2:d:10.1007_s10957-021-01846-7
    DOI: 10.1007/s10957-021-01846-7
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    References listed on IDEAS

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    1. Jiawei Chen & Jun Li & Xiaobing Li & Yibing Lv & Jen-Chih Yao, 2020. "Radius of Robust Feasibility of System of Convex Inequalities with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 384-399, February.
    2. Emilio Carrizosa & Stefan Nickel, 2003. "Robust facility location," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 331-349, November.
    3. T. D. Chuong & V. Jeyakumar, 2017. "An Exact Formula for Radius of Robust Feasibility of Uncertain Linear Programs," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 203-226, April.
    4. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    5. Goberna, M.A. & Jeyakumar, V. & Li, G. & Vicente-Pérez, J., 2015. "Robust solutions to multi-objective linear programs with uncertain data," European Journal of Operational Research, Elsevier, vol. 242(3), pages 730-743.
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    1. Goberna, M.A. & Jeyakumar, V. & Li, G. & Vicente-Pérez, J., 2022. "The radius of robust feasibility of uncertain mathematical programs: A Survey and recent developments," European Journal of Operational Research, Elsevier, vol. 296(3), pages 749-763.
    2. Miguel A. Goberna & Vaithilingam Jeyakumar & Guoyin Li, 2024. "The Stability of Robustness for Conic Linear Programs with Uncertain Data," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1509-1530, November.

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