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A Method for Uncertain Linear Optimization Problems Through Polytopic Approximation of the Uncertainty Set

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Abstract

In this work, we propose a globally convergent iterative method to solve uncertain constrained linear optimization problems. Due to the nondeterministic nature of such a problem, we use the min-max approach to convert the given problem into a deterministic one. We show that the robust feasible sets of the problem corresponding to the uncertainty set and the convex hull of the uncertainty set are identical. This result helps to reduce the number of inequality constraints of the problem drastically; often, this result reduces the semi-infinite programming problem of the min-max robust counterpart into a problem with a finite number of constraints. Following this, we provide a necessary and sufficient condition for the boundedness of the robust feasible set of the problem. Moreover, we explicitly identify the robust feasible set of the problem for polytopic and ellipsoidal uncertainty sets. We present an algorithm to construct an inner polytope of the convex hull of a general uncertainty set under a certain assumption. This algorithm provides a point-wise inner polytopic approximation of the convex hull with arbitrarily small precision. We employ this inner polytopic approximation corresponding to the uncertainty set and the infeasible interior-point technique to derive an iterative approach to solve general uncertain constrained linear optimization problems. Global convergence for the proposed method is reported. Numerical experiments illustrate the practical behaviour of the proposed method on discrete, star-shaped, disc-shaped, and ellipsoidal uncertainty sets.

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Acknowledgements

The authors are thankful to two anonymous reviewers for their critical comments to improve the paper.

Funding

Debdas Ghosh acknowledges the financial support of the research grants MATRICS (MTR/2021/000696) and Core Research Grant (CRG/2022/001347) by the Science and Engineering Research Board, India. Ravi Raushan thankfully acknowledges financial support from CSIR, India, through a research fellowship (File No. 09/1217(13822)/2022-EMR-I) to carry out this research work. Yong Zhao was supported in part by the National Natural Science Foundation of China under grants (12001072), the NSFC-RGC (Hong Kong) Joint Research Program (12261160365), the Chongqing Natural Science Foundation Project under grant (CSTB2022NSCQMSX1318) and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202400760).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, 221005, India

    Ravi Raushan & Debdas Ghosh

  2. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Yong Zhao

  3. Hebei Key Laboratory of Machine Learning and Computational Intelligence and College of Mathematics and Information Science, Hebei University, Baoding, 071002, People’s Republic of China

    Zhou Wei

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  1. Ravi Raushan
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Correspondence to Yong Zhao.

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Raushan, R., Ghosh, D., Zhao, Y. et al. A Method for Uncertain Linear Optimization Problems Through Polytopic Approximation of the Uncertainty Set. J Optim Theory Appl 205, 14 (2025). https://doi.org/10.1007/s10957-025-02626-3

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