ALTERNATING CQ-ALGORITHMS FOR CONVEX FEASIBILITY AND SPLIT FIXED-POINT PROBLEMS

作者：Moudafi, A.^*

来源：Journal of Nonlinear and Convex Analysis, 2014, 15(4): 809-818.

• Feasibility problem
• split fixed-point problem
• CQ-algorithm
• alternating algorithm
• firmly quasi-nonexpansive operator

Let H-1, H-2, H-3 be real Hilbert spaces, let C subset of H-1, Q subset of H-2 be two nonempty closed convex sets, let A : H-1 -> H-3, B : H-2 -> H-3 be two bounded linear operators. We first consider the following new convex feasibility problem @@@ (SEP) Find x is an element of C, y is an element of Q such that Ax = By. @@@ Given a sequence (gamma(k)) of positive parameters and two initial arbitrarily points x(0) is an element of H-1 and y(0) is an element of H-2, we then present and study the convergence of the following new alternating CQ-algorithm @@@ (ACQA) x(k+1) = P-C(x(k) - gamma(k)A*(Ax(k) - By(k))); y(k+1) = P-Q(y(k) + gamma B-k* (Ax(k+1) - By(k))), @@@ where A* and B* denote the adjoint operators of A and B respectively. @@@ Note that, by taking B = I, in (1.1), we recover the convex feasibility problem originally introduced in Censor and Elfving [9] and used later in intensity-modulated radiation therapy. If in addition gamma(k) = 1, in (ACQA), we obtain the related CQ-algorithm introduced by Byrne [6] and applied to dynamic emission tomographic image reconstruction. An extension to a new split common fixed-point problem governed by firmly quasi-nonexpansive mappings is presented and some examples are also provided.

• 出版日期2014