  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.

Optimization and Control