Alternating minimization and projection methods for nonconvex problems

摘要: We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions 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, $Q:\R^n\times\R^m\rightarrow \R$ is a smooth $C^1$ function which couples variables $x$ $y$. The can be viewed as regularization usual Gauss-Seidel method to minimize $L$. work in setting, just assuming that $L$ satisfies Kurdyka-Ł ojasiewicz inequality. An entire section illustrates relevancy such assumption by giving examples ranging from semialgebraic geometry "metrically regular" problems. Our main result stated follows: If L has Kurdyka-Łojasiewicz property, then each bounded sequence generated converges critical point This completed rate algorithm, depends on geometrical around its points. When specialized $Q(x,y)=|x-y|^2$ $f$, $g$ indicator projection mehod (a variant Von Neumann's) wide class sets including tame sets, transverse manifolds or with "regular" intersection. In order illustrate our results concrete problems, we provide convergent reweighted $\ell^1$ compressive sensing application rank reduction