Global Convergence of a New Hybrid Gauss–Newton Structured BFGS Method for Nonlinear Least Squares Problems

Zhou, Weijun; Chen, Xiaojun

doi:10.1137/090748470

Cited by 31 publications

(26 citation statements)

References 33 publications

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“…Figures show, NM-SBFGS-Z is competitive and it outperforms the other algorithms with respect to the total number of function and gradient evaluations and the running time. These observations imply that our modified NM-SBFGS-Z algorithm with ξ = 5 3 turns out to be practically effective.…”

Section: Numerical Experimentsmentioning

confidence: 69%

“…Dennis, Martínez and Tapia presented some quasi-Newton methods with partially known Hessian [4]. The structure principle is an important concept of structured quasi-Newton methods for these problems [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…where ξ is a non-negative constant such that ξ ∈ [0, 3] and u k ∈ R n is any vector parameter satisfying s T k u k = 0 and γ k is defined by (1.11). Also, Khoshgam and Ashrafi obtained a special scalar 5 3 for ξ using the fifth order Teylor expansion of the objective function about the iterate x k+1 . They proved thatỹ k is better than y k due toỹ k approximates ∇ 2 f (x k+1 )s k with higher accuracy [9].…”

Section: Introductionmentioning

confidence: 99%

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A new modified structured BFGS algorithm by utilizing a new modified secant equation

2019

J. Nonlinear Funct. Anal.

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In this paper, we consider a modified structured Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm to solve an unconstrained optimization problem with partial information on the Hessian. To approximate the Hessian matrix, we propose a modified secant (quasi-Newton) equation. Under appropriate conditions, we show that the proposed algorithm has local and super-linear convergence. Numerical experiments are done on a set of the unconstrained non-linear least squares problems. The results demonstrate the practical effectiveness of the proposed algorithm.

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Section: Numerical Experimentsmentioning

confidence: 69%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A new modified structured BFGS algorithm by utilizing a new modified secant equation

2019

J. Nonlinear Funct. Anal.

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“…For non-zero residual problems, it converges superlinearly, and quadratically for zero residual problems. This approach was studied in several early papers [1], [9], [11], [12], and later in [22], [23], [24], [31]. The selection criterion may be rewritten to the following form of…”

Section: Hybrid Bfgsmentioning

confidence: 99%

Estimation of the Cartographic Projection and~its Application in Geoinformatics-habilitation thesis presentation

Bayer

2017

Geoinformatics FCE CTU

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Abstract. Modern techniques for the map analysis allow for the creation of full or partial geometric reconstruction of its content. The projection P(ϕ k , λ k , ϕ 1 , λ 0 , κ) is described by the set of estimated constant values: transformed pole position [ϕ k , λ k ], standard parallel ϕ 1 , longitude λ 0 of the central meridian, and constant parameter κ. Analogously the analyzed map M (R , ∆X, ∆Y, α) is represented by its constant values: auxiliary sphere radius R , origin shifts ∆X, ∆Y , and angle of rotation α. Several new methods denoted as M6-M9 for the estimation of an unknown map projection and its parameters differing in the number of determined parameters, reliability, robustness, and convergence have been developed. However, their computational demands are similar. Instead of directly measuring the dissimilarity δ of two projections, the analyzed map M in an unknown projection and the image M of the sphere S 2 in the well-known (i.e., analyzed) projection P x are compared. Several distance functions for the similarity measurements based on the location as well as shape similarity approaches are proposed. An unconstrained global optimization problem poorly scaled, with large residuals, for the vector of unknown parameters x is solved by the hybrid BFGS method. To avoid a slower convergence rate for small residual problems, it has the ability to switch between first-and second-order methods. Such an analysis is beneficial and interesting for historic, old, or current maps without information about the projection. Its importance is primarily referred to refinement of spatial georeference for the medium-and small-scale maps, analysis of the knowledge about the former world, analysis of the incorrectly/inaccurately drawn regions, and appropriate cataloging of maps. The proposed algorithms have been implemented in the new version of the detectproj software.

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“…The details will be presented in section 2.2. Note that a similar idea has been presented in [47] for the unconstrained nonlinear least square problems [24,37]. Then our subproblem at the k-th iteration is constructed as…”

mentioning

confidence: 99%

Structured Quasi-Newton Methods for Optimization with Orthogonality Constraints

Hu¹,

Jiang²,

Lin³

et al. 2019

SIAM J. Sci. Comput.

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In this paper, we study structured quasi-Newton methods for optimization problems with orthogonality constraints. Note that the Riemannian Hessian of the objective function requires both the Euclidean Hessian and the Euclidean gradient. In particular, we are interested in applications that the Euclidean Hessian itself consists of a computational cheap part and a significantly expensive part. Our basic idea is to keep these parts of lower computational costs but substitute those parts of higher computational costs by the limited-memory quasi-Newton update. More specically, the part related to Euclidean gradient and the cheaper parts in the Euclidean Hessian are preserved. The initial quasi-Newton matrix is further constructed from a limited-memory Nyström approximation to the expensive part. Consequently, our subproblems approximate the original objective function in the Euclidean space and preserve the orthogonality constraints without performing the so-called vector transports. When the subproblems are solved to sufficient accuracy, both global and local q-superlinear convergence can be established under mild conditions. Preliminary numerical experiments on the linear eigenvalue problem and the electronic structure calculation show the effectiveness of our method compared with the state-of-art algorithms.

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Global Convergence of a New Hybrid Gauss–Newton Structured BFGS Method for Nonlinear Least Squares Problems

Cited by 31 publications

References 33 publications

A new modified structured BFGS algorithm by utilizing a new modified secant equation

A new modified structured BFGS algorithm by utilizing a new modified secant equation

Estimation of the Cartographic Projection and~its Application in Geoinformatics-habilitation thesis presentation

Structured Quasi-Newton Methods for Optimization with Orthogonality Constraints

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