String-Averaging Projected Subgradient Methods for Constrained Minimization

TL;DR

This paper introduces a novel approach for constrained minimization that replaces full projections with projections onto individual sets, using dynamic string-averaging projection methods to improve efficiency and generalize previous algorithms.

Contribution

It proposes a new algorithmic scheme combining dynamic string-averaging projections with feasibility-seeking methods for constrained minimization problems.

Findings

The method effectively replaces full feasible region projections with projections onto individual sets.

It generalizes earlier projection algorithms with variable strings and weights.

The approach enhances the flexibility and potential efficiency of constrained minimization algorithms.

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.