String-Averaging Projected Subgradient Methods for Constrained Minimization

Abstract: We consider constrained minimization problems and propose to replace the projection onto the entire feasible region, required in the Projected Subgradient Method (PSM), by projections onto the individual sets whose intersection forms the entire feasible region. Specifically, we propose to perform such projections onto the individual sets in an algorithmic regime of a feasibility-seeking iterative projection method. For this purpose we use the recently developed family of Dynamic String-Averaging Projection (DSAP) methods wherein iteration-index-dependent variable strings and variable weights are permitted. This gives rise to an algorithmic scheme that generalizes, from the algorithmic structural point of view, earlier work of Helou Neto and De Pierro, of Nedi\'c, of Nurminski, and of Ram et al.