Comparing the Methods of Alternating and Simultaneous Projections for Two Subspaces

We study the well-known methods of alternating and simultaneous projections when applied to two nonorthogonal linear subspaces of a real Euclidean space. Assuming that both of the methods have a common starting point chosen from either one of the subspaces, we show that the method of alternating projections converges significantly faster than the method of simultaneous projections. On the other hand, we provide examples of subspaces and starting points, where the method of simultaneous projections outperforms the method of alternating projections

An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space

This paper is concerned with the variational inequality problem (VIP) over the fixed point set of a quasi-nonexpansive operator. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen half-space, and prove that the sequences it generates converge to the unique solution of the VIP. We also present an application of our result to a hierarchical optimization problem.Comment: Accepted for publication in Numerical Functional Analysis and Optimizatio