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

Author

Listed:
  • Liang Chen

    (Chinese Academy of Sciences)

  • Yu-Hong Dai

    (Chinese Academy of Sciences
    University of Chinese Academy of Sciences)

  • Zhou Wei

    (Hebei University
    Yunnan University)

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.

Suggested Citation

  • Liang Chen & Yu-Hong Dai & Zhou Wei, 2022. "Sufficient conditions for existence of global minimizers of functions on Hilbert spaces," Journal of Global Optimization, Springer, vol. 84(1), pages 137-147, September.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:1:d:10.1007_s10898-022-01133-3
    DOI: 10.1007/s10898-022-01133-3
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    References listed on IDEAS

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    1. F. Bernard & L. Thibault & D. Zagrodny, 2005. "Integration of Primal Lower Nice Functions in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 561-579, March.
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