Alternating minimization methods for strongly convex optimization

We consider alternating minimization procedures for convex and non-convex optimization problems with the vector of variables divided into several blocks, each block being amenable for minimization with respect to its variables while maintaining other variables blocks constant. In the case of two blocks, we prove a linear convergence rate for an alternating minimization procedure under the Polyak–Łojasiewicz (PL) condition, which can be seen as a relaxation of the strong convexity assumption. Under the strong convexity assumption in the many-blocks setting, we provide an accelerated alternating minimization procedure with linear convergence rate depending on the square root of the condition number as opposed to just the condition number for the non-accelerated method. We also consider the problem of finding an approximate non-negative solution to a linear system of equations A ⁢ x = y {Ax=y} with alternating minimization of Kullback–Leibler (KL) divergence between Ax and y.