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Sufficient conditions for existence of global minimizers of functions on Hilbert spaces

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Abstract

Given an objective function (differentiable or nondifferentiable), an important optimization problem is to find the global minimizer of the given function and a natural issue is to identify such functions with the global property. In this paper, we study real-valued functions defined on a Hilbert space and provide several sufficient conditions to ensure the existence of the global minimizer of such functions. For a proper lower semicontinuous function, we prove that a global minimizer can be guaranteed if it is bounded below, has the primal-lower-nice property and satisfies the generalized Palais-Smale condition. This result can cover the classic differentiable case whose proof depends heavily on the global existence of the ordinary differential equation. Several examples are constructed to show that the global minimizer may be violated if any of three conditions above is dropped.

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Acknowledgements

The authors wish to thank the referees for careful reading of the paper and for many valuable comments, which drew our attention to [8] and helped to improve our presentation.

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Liang Chen

  2. LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yu-Hong Dai

  3. School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yu-Hong Dai

  4. College of Mathematics and Information Science, Hebei University, Baoding, 071002, China

    Zhou Wei

  5. Department of Mathematics, Yunnan University, Kunming, 650091, China

    Zhou Wei

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  1. Liang Chen
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  2. Yu-Hong Dai
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  3. Zhou Wei
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Correspondence to Zhou Wei.

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This research was supported by the National Natural Science Foundations of China (Grant Nos. 11991021, 12021001, 11971422, 11826204 and 11826206). Research of the third author was supported by CAS “Light of West China” Program and by Jointed Key Project of Yunnan Provincial Science and Technology Department and Yunnan University [No. 2018FY001014] and Program for Innovative Research Team (in Science and Technology) in Universities of Yunnan Province [No. C176240111009]

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Chen, L., Dai, YH. & Wei, Z. Sufficient conditions for existence of global minimizers of functions on Hilbert spaces. J Glob Optim 84, 137–147 (2022). https://doi.org/10.1007/s10898-022-01133-3

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  • DOI: https://doi.org/10.1007/s10898-022-01133-3

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