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, 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 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 Nedić, of Nurminski, and of Ram et al.

Keywords:

• fixed point
• Hilbert space
• metric projection
• non-expansive operator
• projected subgradient minimization
• string-averaging projection methods
• variable strings
• variable weights

2010 Mathematics Subject Classification (MSC):

• 65K10
• 90C25

Note added in proof

We became aware of the paper “Properties of A Class of approximately shrinking operators and their applications” by A. Cegielski and R. Zalas [Fixed Point Theory, 2014, to appear] in which an abstract variational inequality in real Hilbert space is studied and for which a generalized hybrid steepest descent method is proposed wherein the string-averaging methodology is used. Although formulated in a more general setting and employing different assumptions in its analysis, this paper is closely related to the results presented here.

Acknowledgements

We greatly appreciate the constructive comments of two anonymous reviewers which helped to improve the paper.

Funding

The work of the first author was partially supported by the United States-Israel Binational Science Foundation (BSF) Grant number 200912 and US Department of Army Award number W81XWH-10-1-0170.