[Paper Review] Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review

Yair Censor, Maroun Zaknoon|arXiv (Cornell University)|Feb 21, 2018

Optimization and Variational Analysis53 references50 citations

TL;DR

A comprehensive review of projection methods for inconsistent convex feasibility problems, summarizing convergence results, algorithms, and key theoretical developments.

ABSTRACT

The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent CFPs exists and is mainly focused on two directions. One is oriented toward defining other solution concepts that will apply, such as proximity function minimization wherein a proximity function measures in some way the total violation of all constraints. The second direction investigates the behavior of algorithms that are designed to solve a consistent CFP when applied to inconsistent problems. This direction is fueled by situations wherein one lacks a priori information about the consistency or inconsistency of the CFP or does not wish to invest computational resources to get hold of such knowledge prior to running his algorithm. In this paper we bring under one roof and telegraphically review some recent works on inconsistent CFPs.

Motivation & Objective

• Summarize the main concepts and motivations behind inconsistent convex feasibility problems (CFP).
• Survey historical and recent projection-based algorithms used to handle inconsistency in CFPs.
• Highlight convergence results and conditions across various projection schemes.
• Bridge connections between theory (fixed-point, variational inequality) and practical applications of projection methods.

Proposed method

• Discuss composition and convergence of projection operators for two sets and extensions to multiple sets.
• Describe cyclic and block-iterative projection schemes and their convergence properties.
• Explain least-squares interpretations and block-Kaczmarz relaxation results in inconsistent settings.
• Summarize product-space reformulations that convert CFPs into equivalent optimization problems.
• Outline Bregman-projection based proximity-function minimization and related algorithmic schemes.
• Present string-averaging projection (SAP) and its convergence in the inconsistent case.

Experimental results

Research questions

• RQ1 What convergence properties do classical projection methods exhibit when the intersection of constraint sets is empty?
• RQ2 Under what conditions do cyclic, block, and string-averaging projection schemes converge, and at what rates?
• RQ3 How do product-space formulations and proximity-function minimization affect solving inconsistent CFPs?
• RQ4 What is the status of De Pierro’s conjecture in the context of underrelaxed projection methods?

Key findings

• Projection methods can converge to fixed points or to least-squares/nearby solutions in inconsistent CFPs under various assumptions.
• Cyclic and block-iterative projections exhibit weak or strong convergence under conditions such as boundedness, finite dimension, or polyhedral sets.
• Underrelaxation and block-Kaczmarz relaxations lead to limits that approach least-squares solutions as the relaxation parameter tends to zero.
• Product-space reformulations enable parallel projection methods to solve inconsistent CFPs and obtain convergence results.
• Bregman-projection based proximity minimization provides a framework where convergence is obtained to a minimizer of a proximity function.
• String-averaging projections extend applicability and retain convergence in the inconsistent case through a unified framework.