An Optimal Lower Bound for Smooth Convex Functions

First order methods endowed with global convergence guarantees operate using global lower bounds on the objective. The tightening of the bounds leads to an increase in theoretical guarantees and in observed practical performance. In this work, we define a global lower bound for smooth objectives that is optimal with respect to the collected oracle information. Our bound can be readily employed by the Gradient Method with Memory to improve its performance. Further using the machinery underlying the optimal bounds, we introduce a modified version of the estimate sequence that we use to construct an Optimized Gradient Method with Memory possessing the best known convergence guarantees for its class of algorithms up to the proportionality constant. We additionally equip the method with an adaptive convergence guarantee adjustment procedure that is an effective replacement for line-search. Simulation results on synthetic but otherwise difficult smooth problems validate the theoretical properties of the bound and of the proposed methods.