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# The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem

SIAM Journal on Optimization, no. 1 (1997): 26-33

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摘要

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 ...更多

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简介

- In this paper the authors consider the Barzilai and Borwein gradient method for the large scale unconstrained minimization problem (1)

min f (x), x∈Rn where f : Rn → R. - Every iteration of the Barzilai and Borwein method requires only O(n) floating point operations and a gradient evaluation.
- The search direction is always the negative gradient direction, but the choice of steplength is not the classical choice of the steepest descent method.
- Barzilai and Borwein [1] observed that this new choice of steplength required less computational work and greatly speeded up the convergence of the gradient method for quadratics

重点内容

- In this paper we consider the Barzilai and Borwein gradient method for the large scale unconstrained minimization problem (1)

min f (x), x∈Rn where f : Rn → R - The Barzilai and Borwein method can be incorporated in a globalization strategy that preserves the good features of the method and only requires 3n storage locations
- Since the search direction is always the negative gradient direction, it is trivial to ensure that descent directions are generated at every iteration
- This is in sharp contrast to the conjugate gradient methods, for which a very accurate line search has to be performed at every iteration to generate descent directions
- Our numerical experiments seem to indicate that the global Barzilai and Borwein algorithm is competitive and sometimes preferable to recent and well-known implementations of the conjugate gradient method
- To explain the local behavior of the Global Barzilai and Borwein (GBB) method is that the Barzilai and Borwein method, given by (2) and (3), is globally convergent for convex quadratic functions

方法

**Method IT g Time**

GBB P R+

CPU time. if the Hessian is singular at the solution as in problem 11, CONMIN and P R+ clearly out perform GBB.

CONMIN and P R+ out perform GBB in number of iterations, except for problems with a well-conditioned Hessian at the solution, in which case the number of iterations is quite similar.- If the Hessian is singular at the solution as in problem 11, CONMIN and P R+ clearly out perform GBB.
- CONMIN and P R+ out perform GBB in number of iterations, except for problems with a well-conditioned Hessian at the solution, in which case the number of iterations is quite similar.
- In some of those cases, the difference in computing time is remarkable

结论

**Concluding remarks**

The Barzilai and Borwein method can be incorporated in a globalization strategy that preserves the good features of the method and only requires 3n storage locations.- Since the search direction is always the negative gradient direction, it is trivial to ensure that descent directions are generated at every iteration.
- The authors' numerical experiments seem to indicate that the global Barzilai and Borwein algorithm is competitive and sometimes preferable to recent and well-known implementations of the conjugate gradient method.
- To explain the local behavior of the GBB method is that the Barzilai and Borwein method, given by (2) and (3), is globally convergent for convex quadratic functions

- Table1: Test problems
- Table2: Results for GBB, CONMIN, and P R+
- Table3: Number of problems for which a method was a winner

基金

- This research was supported in part by CDCH-UCV project 03.13.0034.93 and by NSF grant CHE9301120

引用论文

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- Concluding remarks. The Barzilai and Borwein method can be incorporated in a globalization strategy that preserves the good features of the method and only requires 3n storage locations. Since the search direction is always the negative gradient direction, it is trivial to ensure that descent directions are generated at every iteration. This is in sharp contrast to the conjugate gradient methods, for which a very accurate line search has to be performed at every iteration to generate descent directions.
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