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

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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.

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Acknowledgements

The author would like to thank the editor and the anonymous referee for their useful comments and valuable suggestions, which helped to improve the paper significantly. This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under Grant Number T2022-18-02.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

    Vo Si Trong Long

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Vo Si Trong Long

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Communicated by Nguyen Mau Nam.

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Long, V.S.T. An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization. J Optim Theory Appl 194, 440–464 (2022). https://doi.org/10.1007/s10957-022-02033-y

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