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In this work we introduce the concept of an Underestimate Sequence (UES),which is motivated by Nesterov's estimate sequence. Our definition of a UESutilizes three sequences, one of which is a lower bound (or under-estimator) ofthe objective function. The question of how to construct an appropriatesequence of lower bounds is addressed, and we present lower bounds for stronglyconvex smooth functions and for strongly convex composite functions, whichadhere to the UES framework. Further, we propose several first order methodsfor minimizing strongly convex functions in both the smooth and compositecases. The algorithms, based on efficiently updating lower bounds on theobjective functions, have natural stopping conditions that provide the userwith a certificate of optimality. Convergence of all algorithms is guaranteedthrough the UES framework, and we show that all presented algorithms convergelinearly, with the accelerated variants enjoying the optimal linear rate ofconvergence.

Fast and Safe: Accelerated gradient methods with optimality certificates and underestimate sequences