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

TLDR

The algorithm is a proximal regularization of the Gauss–Seidel method for minimizing a nonconvex function L, relying on the Kurdyka–Łojasiewicz inequality. The study investigates the convergence behavior of an alternating proximal minimization scheme applied to nonconvex structured functions L(x,y)=f(x)+Q(x,y)+g(y) with proper lower semicontinuous f and g and smooth coupling Q. The method alternates proximal updates on x and y for L, with f and g being proper lower semicontinuous and Q smooth, and is illustrated by examples from semialgebraic geometry, metrically regular problems, and concrete algorithms such as a convergent proximal reweighted ℓ¹ scheme for compressive sensing and a rank‑reduction procedure. If L satisfies the Kurdyka–Łojasiewicz property, every bounded sequence generated by the algorithm converges to a critical point of L, with a convergence rate governed by the geometry near critical points, and the scheme specializes to an alternating projection method that converges for broad classes of sets, including semialgebraic,.

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 and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that 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-Łojasiewicz 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-Łojasiewicz 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 [Formula: see text] 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. 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.