Quantitative analysis of a subgradient-type method for equilibrium problems

TL;DR

This paper applies proof mining techniques to derive explicit convergence rates for a subgradient algorithm solving equilibrium problems in convex optimization, building on prior non-effective convergence proofs.

Contribution

It provides the first explicit rates of metastability and convergence for a subgradient-type method in equilibrium problems using logic-based analysis.

Findings

Derived explicit convergence rates under metric regularity

Applied proof mining to analyze Fejér monotonicity

Extended the analysis to fixed-point sets of nonexpansive mappings

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 Fej\'er monotonicity as given by the second author in previous papers.