Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections

2024/01/30 by Ochiai, Hiroyuki, Sekiguchi, Yoshiyuki, Waki, Hayato
#41A25 #FOS: Mathematics #Optimization and Control (math.OC) #Primary 90C25 #Secondary 65K10

paper · doi:10.48550/arxiv.2401.16689

Abstract

We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to give the convergence rate of alternating projections for the positive semidefinite cone and a line. As a consequence, we show that such alternating projections converge at O(k^-1/2), independently of the singularity degree. A sufficient condition for the linear convergence is also obtained. Our method directly analyzes the defining equation for an alternating projection sequence and does not use error bounds.

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