On the convergence of the block linear Gauss–Seidel method under convex constraints

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 component-wise 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.