Journal ArticleDOI
The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem
TLDR
Results indicate that the global Barzilai and Borwein method may allow some significant reduction in the number of line searches and also in theNumber of gradient evaluations.Abstract:
The Barzilai and Borwein gradient method for the solution of large scale unconstrained minimization problems is considered. This method requires few storage locations and very inexpensive computations. Furthermore, it does not guarantee descent in the objective function and no line search is required. Recently, the global convergence for the convex quadratic case has been established. However, for the nonquadratic case, the method needs to be incorporated in a globalization scheme. In this work, a nonmonotone line search strategy that guarantees global convergence is combined with the Barzilai and Borwein method. This strategy is based on the nonmonotone line search technique proposed by Grippo, Lampariello, and Lucidi [SIAM J. Numer. Anal., 23 (1986), pp. 707--716]. Numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented. These results indicate that the global Barzilai and Borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations.read more
Citations
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Journal ArticleDOI
Global Convergence of a Spectral Conjugate Gradient Method for Unconstrained Optimization
Jinkui Liu,Youyi Jiang +1 more
TL;DR: In this article, a nonlinear spectral conjugate descent method for solving unconstrained optimization problems is proposed on the basis of the CD method and the spectral convolutional gradient method.
Journal ArticleDOI
A modified two-point stepsize gradient algorithm for unconstrained minimization
TL;DR: Based on a modified secant equation proposed by Li and Fukushima, a stepsize is derived for the Barzilai–Borwein gradient method and the limit point of the sequence generated by the algorithm is first-order critical.
Journal ArticleDOI
A gradient method for unconstrained optimization in noisy environment
TL;DR: A gradient method for solving unconstrained minimization problems in noisy environment with almost sure convergence due to a finite number of line-search steps followed by infinitely many SA consecutive steps is proposed and analyzed.
Posted Content
Rayleigh functional for nonlinear systems
TL;DR: In this article, the extremals of the Rayleigh functional coincide with the stationary solutions of the Euler-Lagrange equation, which gives rise to a powerful numerical optimization method in the search for the energy minimizers.
Journal ArticleDOI
Solving the quadratic trust-region subproblem in a low-memory BFGS framework
TL;DR: A new matrix-free method for the large-scale trust-region subproblem, assuming that the approximate Hessian is updated by the L-BFGS formula with m=1 or 2, which constructs a positive definite matrix whose inverse can be expressed analytically, without using factorization.
References
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Book
Practical Methods of Optimization
TL;DR: The aim of this book is to provide a Discussion of Constrained Optimization and its Applications to Linear Programming and Other Optimization Problems.
Book
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
TL;DR: In this paper, Schnabel proposed a modular system of algorithms for unconstrained minimization and nonlinear equations, based on Newton's method for solving one equation in one unknown convergence of sequences of real numbers.
Book
Numerical methods for unconstrained optimization and nonlinear equations
TL;DR: Newton's Method for Nonlinear Equations and Unconstrained Minimization and methods for solving nonlinear least-squares problems with Special Structure.
Journal ArticleDOI
Two-Point Step Size Gradient Methods
TL;DR: Etude de nouvelles methodes de descente suivant le gradient for the solution approchee du probleme de minimisation sans contrainte. as mentioned in this paper.