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Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods

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Abstract

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem’s subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call “zero-convexity”. This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and non-convex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.

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Acknowledgments

We thank Luba Tetruashvili for many helpful comments. We thank Benar Svaiter for helpful discussions and suggestions, in particular, for his ideas regarding the alternative proof of Proposition 1c mentioned in Remark 4. Thanks are also due to Jefferson Melo, Alfredo Iusem, Simeon Reich, Alex Segal, and Mikhail Solodov for helpful remarks, in particular regarding some of the references, and to Claudia Sagastizábal for a helpful discussion on some general aspects of the paper. Finally, we greatly appreciate the constructive and insightful comments of three anonymous reviewers and the Associate Editor which helped us improve the paper. This research was supported by the United States-Israel Binational Science Foundation (BSF) Grant Number 200912, by the US Department of Army award number W81XWH-10-1-0170, and by a special postdoc fellowship from IMPA.

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  1. Department of Mathematics, University of Haifa, Mount Carmel, 3498838 , Haifa, Israel

    Yair Censor

  2. Instituto de Ciências Matemáticas e de Computação (ICMC), University of São Paulo, Avenida Trabalhador São-carlense, 400 - Centro, São Carlos, SP, CEP: 13566-590, Brazil

    Daniel Reem

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  1. Yair Censor
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This work was done while the Daniel Reem was at the Department of Mathematics, University of Haifa, Haifa, Israel (2010–2011), and at IMPA—Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil (2011–2013).

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Censor, Y., Reem, D. Zero-convex functions, perturbation resilience, and subgradient projections for feasibility-seeking methods. Math. Program. 152, 339–380 (2015). https://doi.org/10.1007/s10107-014-0788-7

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