New Hybrid Conjugate Gradient and Broyden–Fletcher–Goldfarb–Shanno Conjugate Gradient Methods

Stanimirović, Predrag S.; Ivanov, Branislav; Djordjević, Snežana; Brajević, Ivona

doi:10.1007/s10957-018-1324-3

Cited by 15 publications

(17 citation statements)

References 24 publications

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“…Further, one of the contemporary trends is to use BFGS update in constructions of new conjugate gradient methods (e.g., see [46]).…”

Section: Resultsmentioning

confidence: 99%

Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods

Djordjević¹

2019

Applied Mathematics

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Here, we consider two important classes of unconstrained optimization methods: conjugate gradient methods and trust region methods. These two classes of methods are very interesting; it seems that they are never out of date. First, we consider conjugate gradient methods. We also illustrate the practical behavior of some conjugate gradient methods. Then, we study trust region methods. Considering these two classes of methods, we analyze some recent results.

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“…Further, one of the contemporary trends is to use BFGS update in constructions of new conjugate gradient methods (e.g., see [46]).…”

Section: Resultsmentioning

confidence: 99%

Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods

Djordjević¹

2019

Applied Mathematics

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“…The hybrid CG-BFGS method has been designed to combine advantages of BFGS and CG methods. It has been shown to reduce the total number of iterations to convergence and is considered here as an alternative to the L-BFGS algorithm. …”

Section: Methodsmentioning

confidence: 99%

Variable-Metric Localization of Occupied and Virtual Orbitals

Luo

Khaliullin

2021

J. Chem. Theory Comput.

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The key idea of the variable-metric approach to orbital localization is to allow nonorthogonality between orbitals while, at the same time, preventing them from becoming linearly dependent. The variable-metric localization has been shown to improve the locality of occupied nonorthogonal orbitals relative to their orthogonal counterparts. In this work, numerous localization algorithms are designed and tested to exploit the conceptual simplicity of the variable-metric approach with the goal of creating a straightforward and reliable localization procedure for virtual orbitals. The implemented algorithms include the steepest descent, conjugate gradient (CG), limited-memory Broyden−Fletcher− Goldfarb−Shanno (L-BFGS), and hybrid procedures as well as trust-region (TR) methods based on the CG and Cauchy-point subproblem solvers. Comparative analysis shows that the CG-based TR algorithm is the best overall method to obtain nonorthogonal localized molecular orbitals (NLMOs), occupied or virtual. The L-BFGS and CG algorithms can also be used to obtain NLMOs reliably but often at higher computational cost. Extensive tests demonstrate that the implemented methods allow us to obtain well-localized Boys−Foster (i.e., maximally localized Wannier functions) and Pipek−Mezey, orthogonal and nonorthogonal, and occupied and virtual orbitals for a variety of gas-phase molecules and periodic materials. The tests also show that virtual NLMOs, which have not been described before, are, on average, 13% (Boys− Foster) and 18% (Pipek−Mezey) more localized than their orthogonal counterparts.

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“…Conjugate gradient methods [3,9,17,22] are very efficient methods for solving large-scale unconstrained optimization problems mainly due to their simplicity and low storage requirements. This has over the years influenced researchers to propose conjugate gradient-based projection methods by combining conjugate gradient methods with the hyperplane projection method [16] to solve nonlinear monotone equations.…”

Section: Introductionmentioning

confidence: 99%

A New Three-Term Conjugate Gradient-Based Projection Method for Solving Large-Scale Nonlinear Monotone Equations

Koorapetse

Kaelo

2019

Mathematical Modelling and Analysis

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A new three-term conjugate gradient-based projection method is presented in this paper for solving large-scale nonlinear monotone equations. This method is derivative-free and it is suitable for solving large-scale nonlinear monotone equations due to its lower storage requirements. The method satisfies the sufficient descent condition FTkdk ≤ −τ‖Fk‖2, where τ > 0 is a constant, and its global convergence is also established. Numerical results show that the method is efficient and promising.

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New Hybrid Conjugate Gradient and Broyden–Fletcher–Goldfarb–Shanno Conjugate Gradient Methods

Cited by 15 publications

References 24 publications

Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods

Unconstrained Optimization Methods: Conjugate Gradient Methods and Trust-Region Methods

Variable-Metric Localization of Occupied and Virtual Orbitals

A New Three-Term Conjugate Gradient-Based Projection Method for Solving Large-Scale Nonlinear Monotone Equations

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