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Solution Uniqueness of Convex Optimization Problems via the Radial Cone

Author

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  • Jalal Fadili

    (Normandie Université)

  • Tran Thai An Nghia

    (Oakland University)

  • Duy Nhat Phan

    (University of Dayton Research Institute)

Abstract

In this paper, we mainly study solution uniqueness of some convex optimization problems. Our characterizations of solution uniqueness are in terms of the radial cone. This approach allows us to know when a unique solution is a strong solution or even a tilt-stable one without checking second-order information. Consequently, we apply our theory to low-rank optimization problems. The radial cone is fully calculated in this case and numerical experiments show that our characterizations are sharp.

Suggested Citation

  • Jalal Fadili & Tran Thai An Nghia & Duy Nhat Phan, 2025. "Solution Uniqueness of Convex Optimization Problems via the Radial Cone," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-32, August.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02723-3
    DOI: 10.1007/s10957-025-02723-3
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    References listed on IDEAS

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    1. Jean Charles Gilbert, 2017. "On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 70-101, January.
    2. Hui Zhang & Wotao Yin & Lizhi Cheng, 2015. "Necessary and Sufficient Conditions of Solution Uniqueness in 1-Norm Minimization," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 109-122, January.
    3. Jérôme Bolte & Shoham Sabach & Marc Teboulle, 2018. "Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1210-1232, November.
    4. Tim Hoheisel & Elliot Paquette, 2023. "Uniqueness in Nuclear Norm Minimization: Flatness of the Nuclear Norm Sphere and Simultaneous Polarization," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 252-276, April.
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