Alternating minimization and projection methods for structured nonconvex problems

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 → R∪{+∞} and g : R → R∪{+∞} are proper lower semicontinuous functions, and Q : R × R → R is a smooth C 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 KurdykaLojasiewicz 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: Assume that L has the KurdykaLojasiewicz property. Then either ‖(xk, yk)‖ → ∞, or the sequence (xk, yk)k∈N 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 an 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. Some illustrative applications to compressive sensing and rank reduction are given.