Convergence Rate Analysis of a Dykstra-Type Projection Algorithm

Abstract

SIAM Journal on Optimization, Volume 34, Issue 1, Page 563-589, March 2024.
Abstract. Given closed convex sets [math], [math], and some nonzero linear maps [math], [math], of suitable dimensions, the multiset split feasibility problem aims at finding a point in [math] based on computing projections onto [math] and multiplications by [math] and [math]. In this paper, we consider the associated best approximation problem, i.e., the problem of computing projections onto [math]; we refer to this problem as the best approximation problem in multiset split feasibility settings (BA-MSF). We adapt the Dykstra’s projection algorithm, which is classical for solving the BA-MSF in the special case when all [math], to solve the general BA-MSF. Our Dykstra-type projection algorithm is derived by applying (proximal) coordinate gradient descent to the Lagrange dual problem, and it only requires computing projections onto [math] and multiplications by [math] and [math] in each iteration. Under a standard relative interior condition and a genericity assumption on the point we need to project, we show that the dual objective satisfies the Kurdyka-Łojasiewicz property with an explicitly computable exponent on a neighborhood of the (typically unbounded) dual solution set when each [math] is [math]-cone reducible for some [math]: this class of sets covers the class of [math]-cone reducible sets, which include all polyhedrons, second-order cone, and the cone of positive semidefinite matrices as special cases. Using this, explicit convergence rate (linear or sublinear) of the sequence generated by the Dykstra-type projection algorithm is derived. Concrete examples are constructed to illustrate the necessity of some of our assumptions.