2024/01/04 by Hiroyuki Ochiai, Ochiai, Hiroyuki, Yoshiyuki Sekiguchi +3
Computer Science · Engineering · Mathematics · #90C25 #Advanced Numerical Analysis Techniques #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Primary 41A25 #Secondary 65K10
paper · doi:10.48550/arxiv.2401.02084
openalex publication_date 2024/01/04 · openalex created_date 2024/01/10 · openalex updated_date 2026/07/28
We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is O(k-1/2), where k is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example.