A generalized subgradient method with relaxation step

We study conditions for convergence of a generalized subgradient algorithm in which a relaxation step is taken in a direction, which is a convex combination of possibly all previously generated subgradients. A simple condition for convergence is given and conditions that guarantee a linear convergence rate are also presented. We show that choosing the steplength parameter and convex combination of subgradients in a certain sense optimally is equivalent to solving a minimum norm quadratic programming problem. It is also shown that if the direction is restricted to be a convex combination of the current subgradient and the previous direction, then an optimal choice of stepsize and direction is equivalent to the Camerini—Fratta—Maffioli modification of the subgradient method.Research supported by the Swedish Research Council for Engineering Sciences (TFR).