The geometric mean algorithm

Ralha, Rui

doi:10.1016/j.amc.2012.08.002

Cited by 2 publications

(1 citation statement)

References 18 publications

(20 reference statements)

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“…The application of the bisection method on the logarithmic scale seems to be an often forgotten technique within the different communities that make use of numerical solvers, and it is certainly under-represented in the literature. We surveyed popular numerical analysis and optimization textbooks, including Boyd and Vandenberghe [3], Chapra and Canale [6], Gill et al [7], Luenberger and Ye [8], Press et al [10], and found no reference to this technique, despite the existence of scattered references in computational forums 2 and other isolated references to "geometric bisection" in the context of eigenvalue computation [11,12]. In fact, it is easy to find textbook examples that recommend the use of relative error stopping criteria in conjunction with bisection method on a linear scale (see pseudo-code in Figure 5.11 of [6] and chapter 9.1 of [10]).…”

Section: Geometric Bisection Fast Trackingmentioning

confidence: 99%

Efficient solvers for Armijo's backtracking problem

David¹,

Caldeira²

2021

Preprint

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Backtracking is an inexact line search procedure that selects the first value in a sequence 𝑥 0 , 𝑥 0 𝛽, 𝑥 0 𝛽 2 ... that satisfies 𝑔(𝑥) ≤ 0 on R + with 𝑔(𝑥) ≤ 0 iff 𝑥 ≤ 𝑥 * . This procedure is widely used in descent direction optimization algorithms with Armijo-type conditions. It both returns an estimate in (𝛽𝑥 * , 𝑥 * ] and enjoys an upper-bound ⌈log 𝛽 𝜖/𝑥 0 ⌉ on the number of function evaluations to terminate, with 𝜖 a lower bound on 𝑥 * . The basic bracketing mechanism employed in several root-searching methods is adapted here for the purpose of performing inexact line searches, leading to a new class of inexact line search procedures. The traditional bisection algorithm for root-searching is transposed into a very simple method that completes the same inexact line search in at most ⌈log 2 log 𝛽 𝜖/𝑥 0 ⌉ function evaluations. A recent bracketing algorithm for root-searching which presents both minmax function evaluation cost (as the bisection algorithm) and superlinear convergence is also transposed, asymptotically requiring ∼ log log log 𝜖/𝑥 0 function evaluations for sufficiently smooth functions. Other bracketing algorithms for root-searching can be adapted in the same way. Numerical experiments suggest time savings of 50% to 80% in each call to the inexact search procedure.CCS Concepts: • Mathematics of computing → Solvers; Numerical analysis.

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