Quantitative analysis of a subgradient-type method for equilibrium problems

Abstract

We use techniques originating from the subdiscipline of mathematical logic called ‘proof mining’ to provide rates of metastability and—under a metric regularity assumption—rates of convergence for a subgradient-type algorithm solving the equilibrium problem in convex optimization over fixed-point sets of firmly nonexpansive mappings. The algorithm is due to H. Iiduka and I. Yamada who in 2009 gave a noneffective proof of its convergence. This case study illustrates the applicability of the logic-based abstract quantitative analysis of general forms of Fejér monotonicity as given by the second author in previous papers.

A computational study of a class of recursive inequalities

“…As such, the content of this section should be seen as a new contribution to applied proof theory in nonlinear analysis, but we aim to present our work without assuming any technical background in the area of application. Gradient descent algorithms have been the subject of case studies in applied proof theory before, notably rates of metastability for a hybrid steepest descent method was given by Körnlein [36] (which also appears as Chapter 8 of Körnlein's thesis [35]), who analyses a theorem of Yamada [58], and a quantitative analysis of a subgradient-like algorithm was recently given by Kohlenbach and Pischke [48], who extract computational information from a theorem of Iiduke and Yamada [18]. Our contribution here, and in particular the concrete application on projective methods in Section 4.2.1, is separate to these (Körnlein considers a different algorithm, while Kohlenbach and Pischke work in R n ), but it would nevertheless be interesting to explore any parallels between these approaches in future work.…”

Quantitative results on algorithms for zeros of differences of monotone operators in Hilbert space

“…López-Acedo and Nicolae [12] on the finitary content of convergence of (quasi-)Fejér monotone sequences, which has been successfully applied in many other contexts of nonlinear analysis, in particular for the asymptotic regularity of compositions of two mappings [11], the proximal point algorithm in uniformly convex Banach spaces [9] and subgradient methods for equilibrium problems [22].…”

Quantitative analysis of a subgradient-type method for equilibrium problems

“…This paper is a condensed version of the Bachelor thesis[10] of the first author written under the supervision of the 2nd author. Date: August 18, 2020.…”