Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods

Hedy Attouch (1) , Jérôme Bolte (2) , Benar Fux Svaiter (3)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier
2 TSE-R - Toulouse School of Economics
3 IMPA

In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficientdecrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward-backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss-Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.

Mots clés: Semi-algebraic optimization; Tame optimization; Kurdyka-Łojasiewicz inequality; Descent methods; Relative error; Sufficient decrease; Forward-backward splitting; Alternating minimization; Proximal algorithms; Iterative thresholding; Block-coordinate methods; o-minimal structures; Nonconvex nonsmooth optimization

Domaines: Optimisation et contrôle [math.OC]

Hedy Attouch, Jérôme Bolte, Benar Fux Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Mathematical Programming, Series A, 2011, 137 (1), pp.91-124. ⟨10.1007/s10107-011-0484-9⟩. ⟨hal-00790042⟩