Alternating minimization and projection methods for nonconvex problems

Details Alternating minimization and projection methods for nonconvex problems Alternating minimization and projection methods for nonconvex problems

2008-01-11

arxiv.org/abs/0801.1780v2

Mathematics

Optimization and Control

Scientific paper

We introduce and study the convergence properties of alternating proximal minimization algorithms for nonconvex and nonsmooth functions. Alternating projection algorithms on closed sets are particular cases of this problem. The novelty of our approach is twofold: first, we work in a nonconvex setting, just assuming that the function under consideration 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. Secondly, we rely on a new class of alternating minimization algorithms with "costs to move" which has recently been introduced by Attouch, Redont and Soubeyran. Our main result can be stated as follows: Assume that objective function has the Kurdyka-Lojasiewicz property and that the sequence is bounded. Then the trajectory has a finite length and, as a consequence, converges to a critical point of the function. 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 (namely the Lojasiewicz exponent). As a striking application, we obtain the convergence of our alternating projection algorithm (a variant of the von Neumann algorithm) for a wide class of sets including in particular semialgebraic and tame sets, transverse smooth manifolds or sets with "regular" intersection.

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