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
The steepest descent algorithm without line search for p-Laplacian
Guangming Zhou,Chunsheng Feng +1 more
TL;DR: Fast convergence of these new algorithms is displayed by their step length figures, which show that if search direction is the steepest descent one, exact step lengths can be substituted properly with step lengths obtained by the formula.
Journal ArticleDOI
En enhanced matrix-free method via double step length approach for solving systems of nonlinear equations
TL;DR: The proposed derivative-free method without computing the Jacobian via acceleration parameter as well as inexact line search procedure is suggested, proven to be globally convergent under mild condition.
Posted Content
An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints
TL;DR: In this article, an augmented Lagrangian method is proposed for solving problems with geometric constraints, such as complementarity and cardinality constraints, which allows for a fast computation of projections onto a nonconvex set of matrices.
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On the acceleration of the Barzilai-Borwein method
TL;DR: A new stepsize to accelerate the Barzilai-Borwein gradient method by requiring finite termination for minimizing two-dimensional strongly convex quadratic function is proposed and gradient methods which adaptively take the nonmonotone BB stepsizes and certain monotone stepsizes for minimizing general strongly conveX quadratics function are developed.
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.