A modified nonmonotone BFGS algorithm for unconstrained optimization

Li, Xiangrong; Wang, Bopeng; Hu, Wujie

doi:10.1186/s13660-017-1453-5

Cited by 9 publications

(3 citation statements)

References 63 publications

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“…rough this formula, the researchers proved that the modified symmetric matrix is positive definite [10]. Case 2: if s T k y k > 0, in this case, we can say surely that the BFGS update matrix is symmetric and positive definite when applied within this formula (in other words, when applying the inequality s T k y k > 0 in the formula h k , max � 0) [11].…”

Section: A New Scalar Formula For the Parameter θ New Kmentioning

confidence: 97%

New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization

Hamed

Al-Kawaz²,

Al-Bayati³

2020

Journal of Mathematics

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This article considers modified formulas for the standard conjugate gradient (CG) technique that is planned by Li and Fukushima. A new scalar parameter θkNew for this CG technique of unconstrained optimization is planned. The descent condition and global convergent property are established below using strong Wolfe conditions. Our numerical experiments show that the new proposed algorithms are more stable and economic as compared to some well-known standard CG methods.

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Section: A New Scalar Formula For the Parameter θ New Kmentioning

confidence: 97%

New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization

Hamed

Al-Kawaz²,

Al-Bayati³

2020

Journal of Mathematics

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“…Unconstrained optimization is the aim of many papers, and it has a variety of applications (see, for example, [1][2][3][4][5]). Nevertheless, the existing numerical methods for solving the general unconstrained optimization problem up to the second order have a very low convergence rate in the case of degenerate problems [6][7][8][9][10][11][12][13][14][15][16][17] since for increasing the convergence rate, it is necessary to use derivatives of orders greater than two [6,7]. At the same time, using derivatives of the 3 rd and 4 th orders makes a numerical method very time-consuming.…”

Section: Introductionmentioning

confidence: 99%

Higher-Order Optimality Conditions for Degenerate Unconstrained Optimization Problems

Zadachyn

2022

JODEA

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In this paper necessary and sufficient conditions of a minimum for the unconstrained degenerate optimization problem are presented. These conditions generalize the well-known optimality conditions. The new optimality conditions are presented in terms of polylinear forms and Hesse's pseudoinverse matrix. The results are illustrated by examples.The formulation and appearance of these conditions differ from high-order optimality conditions by other authors. The suggested representation of high-order optimality conditions makes them convenient for the evaluation of the convergence rate for unconstrained optimization methods in the case of a singular minimum point, for example, for the analysis of Newton's and quasi-Newton's methods.

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“…However, the method converges for only uniformly convex functions. Li et al [25] proposed a new BFGS algorithm with modified secant equation which achieves both global and superlinear convergence for generally convex functions under the nonmonotone line search of [19]. Su and Rong [26] introduced and established a new spectral CG method and its implementation under a modified nonmonotone line search technique.…”

Section: Introductionmentioning

confidence: 99%

A Scaled Conjugate Gradient Method Based on New BFGS Secant Equation with Modified Nonmonotone Line Search

Woldu¹,

Zhang²,

Fissuh³

2020

AJCM

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In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles. ς ρ < < < . CG methods use relatively little memory for large scale problems and require no numerical linear algebra, so each step is quite fast. However, they do not have second order information of the objective function, and typically converge much more slowly than Newton or quasi-Newton methods.The quasi-Newton method is an iterative method with second order information of the objective function, and BFGS is the effective quasi-Newton method T. G. Woldu et al.

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A modified nonmonotone BFGS algorithm for unconstrained optimization

Cited by 9 publications

References 63 publications

New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization

New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization

Higher-Order Optimality Conditions for Degenerate Unconstrained Optimization Problems

A Scaled Conjugate Gradient Method Based on New BFGS Secant Equation with Modified Nonmonotone Line Search

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