Convergence of the projected surrogate constraints method for the linear split feasibility problems

The surrogate constraints method (SC-method) for linear feasibility problems (LFP) is an important tool in convex optimization, especially in large scale optimization. The classical version of the SC-method converges to a solution if the LFP is feasible [17]. Unfortunately, in applications the LFP is often infeasible. Such a situation occurs in computer tomography and in intensity modulated radiation therapy which can be modelled as LFP [5, 11, 13, 16]. In this case one can apply the simultaneous projection method (SP-method) [2, 4, 15] which is actually a short step version of a special case of the SC-method [7]. The SP-method converges to a solution if the LFP is feasible and to an approximate solution in other case. Because of long steps, the SC-method converges faster than the SP-method if the LFP is feasible. Unfortunately, the SC-method diverges if the problem is infeasible. We deal in the paper with the linear split feasibility problems (LSFP) which are more general than the LFP. We analyze the convergence of various versions of the projected surrogate constraints method (PSC-method) for the LSFP in dependence on the step size and on the choice of weights. We show also that the convergence of the SC-method of Yang-Murty [17] and of the CQ-method of Byrne [5] applied to the LSFP follows from our main result.