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We study the convergence properties of an alternating proximal minimizationalgorithm for nonconvex structured functions of the type:$L(x,y)=f(x)+Q(x,y)+g(y)$, where $f:\R^n\rightarrow\R\cup{+\infty}$ and$g:\R^m\rightarrow\R\cup{+\infty}$ are proper lower semicontinuous functions,and $Q:\R^n\times\R^m\rightarrow \R$ is a smooth $C^1$ function which couplesthe variables $x$ and $y$. The algorithm can be viewed as a proximalregularization of the usual Gauss-Seidel method to minimize $L$. We work in anonconvex setting, just assuming that the function $L$ satisfies the Kurdyka-\Lojasiewicz inequality. An entire section illustrates the relevancy of such anassumption by giving examples ranging from semialgebraic geometry to"metrically regular" problems. Our main result can be stated as follows: If Lhas the Kurdyka-\L ojasiewicz property, then each bounded sequence generated bythe algorithm converges to a critical point of $L$. This result is completed bythe study of the convergence rate of the algorithm, which depends on thegeometrical properties of the function $L$ around its critical points. Whenspecialized to $Q(x,y)=|x-y|^2$ and to $f$, $g$ indicator functions, thealgorithm is an alternating projection mehod (a variant of Von Neumann's) thatconverges for a wide class of sets including semialgebraic and tame sets,transverse smooth manifolds or sets with "regular" intersection. In order toillustrate our results with concrete problems, we provide a convergent proximalreweighted $\ell^1$ algorithm for compressive sensing and an application torank reduction problems.

Proximal alternating minimization and projection methods for nonconvex problems. An approach based on the Kurdyka-Lojasiewicz inequality