A Line-Search Algorithm Inspired by the Adaptive Cubic Regularization Framework and Complexity Analysis

El Houcine Bergou (), Youssef Diouane () and Serge Gratton ()
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El Houcine Bergou: Université Paris-Saclay
Youssef Diouane: Université de Toulouse
Serge Gratton: Université de Toulouse

Journal of Optimization Theory and Applications, 2018, vol. 178, issue 3, No 9, 885-913

Abstract: Abstract Adaptive regularized framework using cubics has emerged as an alternative to line-search and trust-region algorithms for smooth nonconvex optimization, with an optimal complexity among second-order methods. In this paper, we propose and analyze the use of an iteration dependent scaled norm in the adaptive regularized framework using cubics. Within such a scaled norm, the obtained method behaves as a line-search algorithm along the quasi-Newton direction with a special backtracking strategy. Under appropriate assumptions, the new algorithm enjoys the same convergence and complexity properties as adaptive regularized algorithm using cubics. The complexity for finding an approximate first-order stationary point can be improved to be optimal whenever a second-order version of the proposed algorithm is regarded. In a similar way, using the same scaled norm to define the trust-region neighborhood, we show that the trust-region algorithm behaves as a line-search algorithm. The good potential of the obtained algorithms is shown 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; 49M05; 49M15; 90C06; 90C60 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-018-1341-2

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