Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates

Pierre Frankel (), Guillaume Garrigos () and Juan Peypouquet ()
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Pierre Frankel: Université Montpellier 2
Guillaume Garrigos: Université Montpellier 2
Juan Peypouquet: Universidad Técnica Federico Santa María

Journal of Optimization Theory and Applications, 2015, vol. 165, issue 3, No 11, 874-900

Abstract: Abstract We study the convergence of general descent methods applied to a lower semi-continuous and nonconvex function, which satisfies the Kurdyka–Łojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of the function, and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward–backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg–Marquardt algorithm is detailed.

Keywords: Nonconvex and nonsmooth optimization; Kurdyka–Łojasiewicz inequality; Descent methods; Convergence rates; Variable metric; Gauss–Seidel method; Newton-like method; 49M37; 65K10; 90C26; 90C30 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (21)

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DOI: 10.1007/s10957-014-0642-3

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