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Two Optimal Value Functions in Parametric Conic Linear Programming

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Abstract

We consider the conic linear program given by a closed convex cone in an Euclidean space and a matrix, where vector on the right-hand side of the inequality constraint and the vector defining the objective function are subject to change. Using the strict feasibility condition, we prove the locally Lipschitz continuity and obtain some differentiability properties of the optimal value function of the problem under right-hand-side perturbations. For the optimal value function under linear perturbations of the objective function, similar differentiability properties are obtained under the assumption saying that both primal problem and dual problem are strictly feasible.

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Acknowledgements

This work was supported by National Foundation for Science & Technology Development (Vietnam), Pukyong National University (Busan, Korea), and Vietnam Institute for Advanced Study in Mathematics (VIASM). The first author was partially supported by the Hanoi National University of Education under Grant Number SPHN20-07: “Differential Stability in Conic Linear Programming”. The second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1A2C1008672). The authors are grateful to the handling Associate Editor and the two anonymous referees for their careful readings, many insightful comments, and detailed suggestions which have helped to improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam

    Nguyen Ngoc Luan

  2. Department of Applied Mathematics, Pukyong National University, Busan, Korea

    Do Sang Kim

  3. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, 10307, Vietnam

    Nguyen Dong Yen

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  1. Nguyen Ngoc Luan
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  2. Do Sang Kim
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  3. Nguyen Dong Yen
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Correspondence to Nguyen Dong Yen.

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Communicated by Boris S. Mordukhovich.

Dedicated to Professor Franco Giannessi on the occasion of his 85th birthday.

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Luan, N.N., Kim, D.S. & Yen, N.D. Two Optimal Value Functions in Parametric Conic Linear Programming. J Optim Theory Appl 193, 574–597 (2022). https://doi.org/10.1007/s10957-021-01959-z

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