Splitting methods with variable metric for KL functions

Pierre Frankel (1) , Guillaume Garrigos (1, 2) , Juan Peypouquet (2)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 Departamento de Matematica [Valparaiso]

We study the convergence of general abstract descent methods applied to a lower semicontinuous nonconvex function f that satisfies the Kurdyka-Lojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of f 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 detailled.

Mots clés: Nonconvex and nonsmooth optimization; Kurdyka-Lojasiewicz inequality; Descent methods; Convergence rates; Variable metric; Gauss-Seidel method; Newton-like method

Domaines: Optimisation et contrôle [math.OC]

Pierre Frankel, Guillaume Garrigos, Juan Peypouquet. Splitting methods with variable metric for KL functions. 2013. ⟨hal-00987523v2⟩