A counterexample to De Pierro's conjecture on the convergence of under-relaxed cyclic projections. (2nd January 2019)

• Record Type: Journal Article
• Authors: Cominetti, Roberto; Roshchina, Vera; Williamson, Andrew
• Is Part Of: Optimization. Volume 68:Number 1(2019)
• Page Start: 3
• Page End: 12
• Publication Date: 2019-01-02
• Subjects: Cyclic projections -- under-relaxed projections -- De Pierro conjecture; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1474471
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, de Pierro conjectured that the limit cycles generated by theε -under-relaxed cyclic projection method converge whenε ↓ 0 towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that it is false in general by constructing a system of three compact convex sets inR 3 for which theε -under-relaxed cycles do not converge.