Efficiency of higher-order algorithms for minimizing composite functions

Yassine Nabou () and Ion Necoara ()
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Yassine Nabou: National University of Science and Technology Politehnica Bucharest
Ion Necoara: National University of Science and Technology Politehnica Bucharest

Computational Optimization and Applications, 2024, vol. 87, issue 2, No 5, 473 pages

Abstract: Abstract Composite minimization involves a collection of functions which are aggregated in a nonsmooth manner. It covers, as a particular case, smooth approximation of minimax games, minimization of max-type functions, and simple composite minimization problems, where the objective function has a nonsmooth component. We design a higher-order majorization algorithmic framework for fully composite problems (possibly nonconvex). Our framework replaces each component with a higher-order surrogate such that the corresponding error function has a higher-order Lipschitz continuous derivative. We present convergence guarantees for our method for composite optimization problems with (non)convex and (non)smooth objective function. In particular, we prove stationary point convergence guarantees for general nonconvex (possibly nonsmooth) problems and under Kurdyka–Lojasiewicz (KL) property of the objective function we derive improved rates depending on the KL parameter. For convex (possibly nonsmooth) problems we also provide sublinear convergence rates.

Keywords: Composite optimization; (Non)convex minimization; Higher-order methods; Kurdyka–Lojasiewicz property; Convergence rates (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-023-00533-9

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