Superiorization and Perturbation Resilience of Algorithms: A Continuously Updated Bibliography

This document presents a, (mostly) chronologically ordered, bibliography of scientific publications on the superiorization methodology and perturbation resilience of algorithms which is compiled and continuously updated by us at: http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the beginings of this topic we try to trace the work that has been published about it since its inception. To the best of our knowledge this bibliography represents all available publications on this topic to date, and while the URL is continuously updated we will revise this document and bring it up to date on arXiv approximately once a year. Abstracts of the cited works, and some links and downloadable files of preprints or reprints are available on the above mentioned Internet page. If you know of a related scientific work in any form that should be included here kindly write to me on: [email protected] with full bibliographic details, a DOI if available, and a PDF copy of the work if possible. The Internet page was initiated on March 7, 2015, and has been last updated on March 12, 2020.Comment: Original report: June 13, 2015 contained 41 items. First revision: March 9, 2017 contained 64 items. Second revision: March 8, 2018 contained 76 items. Third revision: March 11, 2019 contains 90 items. Fourth revision: March 16, 2020 contains 112 item

New Douglas-Rachford algorithmic structures and their convergence analyses

In this paper we study new algorithmic structures with Douglas- Rachford (DR) operators to solve convex feasibility problems. We propose to embed the basic two-set-DR algorithmic operator into the String-Averaging Projections (SAP) and into the Block-Iterative Pro- jection (BIP) algorithmic structures, thereby creating new DR algo- rithmic schemes that include the recently proposed cyclic Douglas- Rachford algorithm and the averaged DR algorithm as special cases. We further propose and investigate a new multiple-set-DR algorithmic operator. Convergence of all these algorithmic schemes is studied by using properties of strongly quasi-nonexpansive operators and firmly nonexpansive operators.Comment: SIAM Journal on Optimization, accepted for publicatio

Weak and Strong Superiorization: Between Feasibility-Seeking and Minimization

We review the superiorization methodology, which can be thought of, in some cases, as lying between feasibility-seeking and constrained minimization. It is not quite trying to solve the full fledged constrained minimization problem; rather, the task is to find a feasible point which is superior (with respect to an objective function value) to one returned by a feasibility-seeking only algorithm. We distinguish between two research directions in the superiorization methodology that nourish from the same general principle: Weak superiorization and strong superiorization and clarify their nature.Comment: Revised version. Presented at the Tenth Workshop on Mathematical Modelling of Environmental and Life Sciences Problems, October 16-19, 2014, Constantza, Romania. http://www.ima.ro/workshop/tenth_workshop

A von Neumann Alternating Method for Finding Common Solutions to Variational Inequalities

Modifying von Neumann's alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space.Comment: Nonlinear Analysis Series A: Theory, Methods & Applications, accepted for publicatio

Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

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.Comment: Mathematical Programming Series A, accepted for publicatio