A FENCHEL DUALITY ASPECT OF ITERATIVE I-PROJECTION PROCEDURES

BHASKAR BHATTACHARYA^{1} AND RICHARD L. DYKSTRA^{2}

1Department of Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, U.S.A. 2Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, U.S.A.

(Received May 17, 1995; revised October 3, 1996)

Abstract. In this paper we interpret Dykstra’s iterative procedure for finding an I-projection onto the intersection of closed, convex sets in terms of its Fenchel dual. Seen in terms of its dual formulation, Dykstra’s algorithm is intuitive and can be shown to converge monotonically to the correct solution. Moreover, we show that it is possible to sharply bound the location of the constrained optimal solution.

Key words and phrases: Algorithm, convex sets, Fenchel duality, I-projections, iterative, Kullback-Leibler.