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A line-search algorithm inspired by the adaptive cubic regularization framework, with a worst-case complexity $\mathcal{O}(\epsilon^{-3/2})$

E. Bergou(el-houcine.bergou***at***jouy.inra.fr)
Y. Diouane(youssef.diouane***at***isae.fr)
S. Gratton(serge.gratton***at***enseeiht.fr)

Abstract: Adaptive regularized framework using cubics (ARC) has emerged as an alternative to line-search and trust-region for smooth nonconvex optimization, with an optimal complexity amongst second-order methods. In this paper, we propose and analyze the use of a special (iteration dependent) scaled norm in ARC of the form $\|x\|_M= \sqrt{x^TMx}$ for $x \in \mathbb{R}^n$, where $M$ is a symmetric positive definite matrix satisfying specific secant equation. Within the proposed norm, ARC behaves as a line-search algorithm along the Newton direction, with a special backtracking strategy and acceptability condition. The obtained line-search algorithm enjoys the same convergence and complexity properties as ARC, in particular the complexity for finding approximate first-order stationary points is $\mathcal{O}(\epsilon^{-3/2})$. Furthermore, we have proposed similar analysis when considering the trust-region framework. The good potential of the new line-search algorithms is showed on a set of large scale optimization problems.

Keywords: Nonlinear optimization, unconstrained optimization, line-search methods, adaptive regularized framework using cubics, trust-region methods, worst-case complexity.

Category 1: Nonlinear Optimization (Unconstrained Optimization )

Citation:

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Entry Submitted: 06/16/2017
Entry Accepted: 06/16/2017
Entry Last Modified: 06/16/2017

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