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An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization

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  • Vo Si Trong Long

    (University of Science
    Vietnam National University)

Abstract

An invariant-point theorem and its equivalent formulation are established in which distance functions are replaced by minimal time functions. It is worth emphasizing here that the class of minimal time functions can be interpreted as a general type of directional distance functions recently used to develop new applications in optimization theory. The obtained results are applied in two directions. First, we derive sufficient conditions for the existence of solutions to optimization-related problems without convexity. As an easy corollary, we get a directional Ekeland variational principle. Second, we propose a new type of global error bounds for inequalities which allows us to simultaneously study nonconvex and convex functions. Several examples and comparison remarks are included as well to explain advantages of our results with existing ones in the literature.

Suggested Citation

  • Vo Si Trong Long, 2022. "An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 440-464, August.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:2:d:10.1007_s10957-022-02033-y
    DOI: 10.1007/s10957-022-02033-y
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    References listed on IDEAS

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    1. Nguyen Mau Nam & Maria Cristina Villalobos & Nguyen Thai An, 2012. "Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 768-791, September.
    2. Phan Khanh & Vo Long, 2014. "Invariant-point theorems and existence of solutions to optimization-related problems," Journal of Global Optimization, Springer, vol. 58(3), pages 545-564, March.
    3. Boris Mordukhovich & Nguyen Nam, 2010. "Limiting subgradients of minimal time functions in Banach spaces," Journal of Global Optimization, Springer, vol. 46(4), pages 615-633, April.
    4. Aram V. Arutyunov & Alexey F. Izmailov, 2006. "Directional Stability Theorem and Directional Metric Regularity," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 526-543, August.
    5. Teodor Chelmuş & Marius Durea & Elena-Andreea Florea, 2019. "Directional Pareto Efficiency: Concepts and Optimality Conditions," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 336-365, July.
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