Convergence rate analysis of a Dykstra-type projection algorithm

Abstract

Given closed convex sets Ci, i = 1, . . . , \ell , and some nonzero linear maps Ai, i = 1, . . . , \ell ,of suitable dimensions, the multiset split feasibility problem aims at finding a point in \bigcap \ell i=1 A 1i Cibased on computing projections onto Ci and multiplications by Ai and ATi . In this paper, weconsider the associated best approximation problem, i.e., the problem of computing projections onto\bigcap \ell i=1 A 1i Ci; we refer to this problem as the best approximation problem in multiset split feasibilitysettings (BA-MSF). We adapt the Dykstra's projection algorithm, which is classical for solving theBA-MSF in the special case when all Ai = I, to solve the general BA-MSF. Our Dykstra-typeprojection algorithm is derived by applying (proximal) coordinate gradient descent to the Lagrangedual problem, and it only requires computing projections onto Ci and multiplications by Ai andATi in each iteration. Under a standard relative interior condition and a genericity assumption onthe point we need to project, we show that the dual objective satisfies the Kurdyka-\Lojasiewiczproperty with an explicitly computable exponent on a neighborhood of the (typically unbounded)dual solution set when each Ci is C1,\alpha -cone reducible for some \alpha \in (0, 1]: this class of sets coversthe class of C2-cone reducible sets, which include all polyhedrons, second-order cone, and the coneof positive semidefinite matrices as special cases. Using this, explicit convergence rate (linear orsublinear) of the sequence generated by the Dykstra-type projection algorithm is derived. Concreteexamples are constructed to illustrate the necessity of some of our assumptions.