Affine-invariant contracting-point methods for Convex Optimization

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

No 2020029, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

Abstract: In this paper, we develop new affine-invariant algorithms for solving composite con- vex minimization problems with bounded domain. We present a general framework of Contracting-Point methods, which solve at each iteration an auxiliary subproblem re- stricting the smooth part of the objective function onto contraction of the initial domain. This framework provides us with a systematic way for developing optimization methods of different order, endowed with the global complexity bounds. We show that using an ap- propriate affine-invariant smoothness condition, it is possible to implement one iteration of the Contracting-Point method by one step of the pure tensor method of degree p ≥ 1. The resulting global rate of convergence in functional residual is then O(1/kp), where k is the iteration counter. It is important that all constants in our bounds are affine-invariant. For p = 1, our scheme recovers well-known Frank-Wolfe algorithm, providing it with a new interpretation by a general perspective of tensor methods. Finally, within our frame- work, we present efficient implementation and total complexity analysis of the inexact second-order scheme (p = 2), called Contracting Newton method. It can be seen as a proper implementation of the trust-region idea. Preliminary numerical results confirm its good practical performance both in the number of iterations, and in computational time.

Keywords: Convex Optimization; Frank-Wolfe algorithm; Newton method; Tensor Methods; Global Complexity Bounds (search for similar items in EconPapers)
Pages: 26
Date: 2020-09-17
New Economics Papers: this item is included in nep-cmp
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Citations: View citations in EconPapers (1)

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