Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz In

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 : Rn → R∪{+∞} and g : Rm → R∪{+∞} are proper lower semicontinuous functions, and Q : Rn × Rm → R is a smooth C1 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 geo...

• Type: Journal
• Year: 2010
• Where: MOR
• Authors: Hedy Attouch, Jérôme Bolte, Patrick Redont, Antoine Soubeyran
• Keywords: Algorithms, Minimization Algorithms, MOR 2010, Proximal