An Open Question on Cyclic Relaxation

Abstract

The problem discussed in this note is highly interesting. It is related to several dual iterative methods, such as the methods proposed by Kaczmarz, Hildreth, Agmon, Cryer, Mangasarian, Herman, Lent, Censor, and others. Cast as ‘row-action methods’ these algorithms have been proved as useful tools for solving large convex feasibility problems that arise in medical image reconstruction from projections, in inverse problems in radiation therapy, and in linear programming. The question that we want to answer is how these algorithms behave when the feasible region is empty. It is shown that under certain conditions the primal sequence still converges while the dual sequence {y k } obeys the rule y k =u k +k v, where {u k } is a converging sequence and v is a fixed vector that satisfies A T v=0,v≥0,and,b T v>0. It is conjectured that these properties hold whenever the feasible region is empty. However, the validity of this claim remains an open question.