To cite this version:Hedy Attouch, Jérôme Bolte, Benar Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program., Ser. A, 2011, 137 (1), pp.91-124. <10.1007/s10107-011-0484-9>. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods , 15, 2010; revised: July, 21, 2011 Abstract In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficient-decrease 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-Lojasiewicz 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.2010 Mathematics Subject Classification: 34G25, 47J25, 47J30, 47J35, 49M15, 49M37, 65K15, 90C25, 90C53.

Abstract. We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x, y) = f (x)+Q(x, y)+g(y), where f : R n → R∪{+∞} and g : R m → R∪{+∞} are proper lower semicontinuous functions, and Q : R n × R m → R is a smooth C 1 function which couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L.We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Lojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to "metrically regular" problems.Our main result can be stated as follows: If L has the Kurdyka-Lojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to Q(x, y) = x − y 2 and to f , g indicator functions, the algorithm is an alternating projection mehod (a variant of Von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with "regular "intersection. In order to illustrate our results with concrete problems, we provide a convergent proximal reweighted ℓ 1 algorithm for compressive sensing and an application to rank reduction problems.

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the Lojasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following Lojasiewicz's original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of Lojasiewicz exponents. Apart from the sharp and elliptic cases which yield finite-time or geometric convergence, the decay estimates that are derived are of the type O(k −s), where s ∈ (0, +∞) depends on the flatness of the function.