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On the convergence of the block nonlinear Gauss-Seidel method under convex constraints. (English) Zbl 0955.90128
Summary: We give new convergence results for the block Gauss-Seidel method for problems where the feasible set is the Cartesian product of $m$ closed convex sets, under the assumption that the sequence generated by the method has limit points. We show that the method is globally convergent for $m=2$ and that for $m>2$ convergence can be established both when the objective function $f$ is componentwise strictly quasiconvex with respect to $m-2$ components and when $f$ is pseudoconvex. Finally, we consider a proximal point modification of the method and we state convergence results without any convexity assumption on the objective function.
##### MSC:
 90C30 Nonlinear programming 49M27 Decomposition methods in calculus of variations 65K05 Mathematical programming (numerical methods)