Inexact accelerated high-order proximal-point methods

Yurii Nesterov ()
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Yurii Nesterov: Université catholique de Louvain, LIDAM/CORE, Belgium

No 3252, LIDAM Reprints CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: In this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate O k−4, where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian.We also present new methods with the exact auxiliary search procedure, which have the rate of convergence O k−(3p+1)/2 , where p ≥ 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

Keywords: Convex optimization; Tensor methods; Proximal-point operator; Lower complexity bounds; Optimal methods (search for similar items in EconPapers)
Pages: 27
Date: 2023-01-01
Note: In: Mathematical Programming, 2023, vol. 197(1), p. 1-26
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Persistent link: https://EconPapers.repec.org/RePEc:cor:louvrp:3252

DOI: 10.1007/s10107-021-01727-x

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