Abstract
The circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP), and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.
Highlights
We consider the convex feasibility problem (CFP) consisting of finding a point in the intersection of a finite number of closed convex sets
circumcentered-reflection method (CRM) was shown in [15] to converge to a solution of CFP, and it was proven in [9] that linear convergence is obtained in the presence of an error bound condition
We prove that under error bound conditions separating schemes are available so that MAAP and circumcentered approximate-reflection method (CARM) enjoy linear convergence rates, with the linear rate of CARM being strictly better than MAAP
Summary
We consider the convex feasibility problem (CFP) consisting of finding a point in the intersection of a finite number of closed convex sets. Definition 2.4 Given a closed and convex set K ⊂ Rn, a separating operator for K is a point-to-set mapping S : Rn → P(Rn) satisfying:. For the family of convex sets in Examples 2.6 and 2.7, we get both explicit separating operators complying with Definition 2.4 and closed formulas for projections onto them. We mention that any closed and convex set K can be written as the 0-sublevel set of a convex and even smooth function g, for instance, g(x) = dist(x, K), but in general this is not advantageous, because for this g it holds that ∇g(x) = 2(x – PK (x)), so that PK (x), the exact projection of x onto K , is needed for computing the separating half-space, and nothing has been won. The result follows with the same argument as in Example 2.6, with zk,i, Si, gi substituting for zk, S, g
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Fixed Point Theory and Algorithms for Sciences and Engineering
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
