Fast and safe: accelerated gradient methods with optimality certificates and underestimate sequences

Majid Jahani (), Naga Venkata C. Gudapati (), Chenxin Ma (), Rachael Tappenden () and Martin Takáč ()
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Majid Jahani: Lehigh University
Naga Venkata C. Gudapati: Lehigh University
Chenxin Ma: Lehigh University
Rachael Tappenden: University of Canterbury
Martin Takáč: Lehigh University

Computational Optimization and Applications, 2021, vol. 79, issue 2, No 5, 369-404

Abstract: Abstract In this work we introduce the concept of an Underestimate Sequence (UES), which is motivated by Nesterov’s estimate sequence. Our definition of a UES utilizes three sequences, one of which is a lower bound (or under-estimator) of the objective function. The question of how to construct an appropriate sequence of lower bounds is addressed, and we present lower bounds for strongly convex smooth functions and for strongly convex composite functions, which adhere to the UES framework. Further, we propose several first order methods for minimizing strongly convex functions in both the smooth and composite cases. The algorithms, based on efficiently updating lower bounds on the objective functions, have natural stopping conditions that provide the user with a certificate of optimality. Convergence of all algorithms is guaranteed through the UES framework, and we show that all presented algorithms converge linearly, with the accelerated variants enjoying the optimal linear rate of convergence.

Keywords: Underestimate sequence; Estimate sequence; Quadratic averaging; Lower bounds; Strongly convex; Smooth minimization; Composite minimization; Accelerated algorithms; 90C25; 90C47; 68Q25 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10589-021-00269-4

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