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

Liang Chen (), Yu-Hong Dai () and Zhou Wei ()
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Liang Chen: Chinese Academy of Sciences
Yu-Hong Dai: Chinese Academy of Sciences
Zhou Wei: Hebei University

Journal of Global Optimization, 2022, vol. 84, issue 1, No 6, 137-147

Abstract: 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.

Keywords: Sufficient condition; Global minmizer; Primal-lower-nice; Subdifferential evolution problem; Palais–Smale condition; 49J52; 49J53; 34G25 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-022-01133-3

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