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.

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ADInternational audienceIn this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem