Mathematical Programming

, Volume 27, Issue 2, pp 155–175

# QN-like variable storage conjugate gradients

• A. Buckley
• A. Lenir
Article

## Abstract

Both conjugate gradient and quasi-Newton methods are quite successful at minimizing smooth nonlinear functions of several variables, and each has its advantages. In particular, conjugate gradient methods require much less storage to implement than a quasi-Newton code and therefore find application when storage limitations occur. They are, however, slower, so there have recently been attempts to combine CG and QN algorithms so as to obtain an algorithm with good convergence properties and low storage requirements. One such method is the code CONMIN due to Shanno and Phua; it has proven quite successful but it has one limitation. It has no middle ground, in that it either operates as a quasi-Newton code using O(n2) storage locations, or as a conjugate gradient code using 7n locations, but it cannot take advantage of the not unusual situation where more than 7n locations are available, but a quasi-Newton code requires an excessive amount of storage.

In this paper we present a way of looking at conjugate gradient algorithms which was in fact given by Shanno and Phua but which we carry further, emphasize and clarify. This applies in particular to Beale's 3-term recurrence relation. Using this point of view, we develop a new combined CG-QN algorithm which can use whatever storage is available; CONMIN occurs as a special case. We present numerical results to demonstrate that the new algorithm is never worse than CONMIN and that it is almost always better if even a small amount of extra storage is provided.

### Key words

Minimization Conjugate Gradient Quasi-Newton Variable Storage Reduced Storage

## Preview

### Bibliography

1. [1]
A. Buckley, “A combined conjugate gradient quasi-Newton minimization algorithm”,Mathematical Programming 15 (1978) 200–210.
2. [2]
A. Buckley, “Extending the relationship between the conjugate gradient and BFGS algorithms”,Mathematical Programming 15 (1978) 343–348.
3. [3]
A. Buckley, “Conjugate gradient methods”, in: M.J.D. Powell, ed.,Nonlinear Optimization 1981, Proceedings of the NATO Advanced Research Institute on Nonlinear Optimization (Academic Press, London, 1982) pp. 17–22.Google Scholar
4. [4]
A. Buckley, “A portable package for testing minimization algorithms”, in: John M. Mulvey, ed.,Proceedings of the COAL Conference on Mathematical Programming Software, Boulder, Colorado (Springer, New York, 1982) 226–235.Google Scholar
5. [5]
R. Fletcher and C.M. Reeves, “Function minimization by conjugate gradients”,Computer Journal 7 (1963) 163–168.
6. [6]
R. Fletcher, “AFortran subroutine for minimization by the method of conjugate gradients”, Report R7073, U.K. A.E.R.E., Harwell, England (1972).Google Scholar
7. [7]
L. Nazareth, “A relationship between the BFGS and conjugate gradient algorithms and its implications for new algorithms”,SIAM Journal on Numerical Analysis 16 (1979) 794–800.
8. [8]
J. Nocedal, “Updating quasi-Newton matrices with limited storage”,Mathematics of Computation 35 (1980) 773–782.
9. [9]
S.S. Oren and E. Spedicato, “Optimal conditioning of self-scaling variable metric algorithms”,Mathematical Programming 10 (1976) 70–90.
10. [10]
A. Perry, “A modified conjugate gradient algorithm”, Discussion paper 229, Center for Mathematical Studies in Economics and Management Science, Northwestern University (1976).Google Scholar
11. [11]
M.J.D. Powell, “Restart procedures for the conjugate gradient method”,Mathematical Programming 12 (1977) 241–254.
12. [12]
D.F. Shanno, “Conjugate gradient methods with inexact searches”,Mathematics of Operations Research 3 (1978) 244–256.
13. [13]
D.F. Shanno and K.-H. Phua, “Numerical comparison of several variable metric algorithms”,Journal of Optimization Theory and Applications 25 (1978) 507–518.
14. [14]
D.F. Shanno, “Remark on Algorithm 500”,ACM Transactions on Mathematical Sofware 6 (1980) 618–622.
15. [15]
Ph. Toint, “Some numerical results using a sparse matrix updating formula in unconstrained optimization”,Mathematics of Computation 32 (1978) 839–851.