Abstract
The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength which satisfies certain standard conditions. Prototype algorithms are described which guarantee finding such a step in a finite number of operations. This is achieved by first bracketing an interval of acceptable values and then reducing this bracket uniformly by the repeated use of sectioning in a systematic way. Some new theorems about convergence and termination of the line search are presented. Use of these algorithms to solve the line search subproblem in methods for nonlinear least squares is considered. We show that substantial gains in efficiency can be made by making polynomial interpolations to the individual residual functions rather than the overall objective function. We also study modified schemes in which the Jacobian matrix is evaluated as infrequently as possible, and show that further worthwhile savings can be made. Numerical results are presented.
| Original language | English |
|---|---|
| Pages (from-to) | 359-377 |
| Number of pages | 19 |
| Journal | Journal of Optimization Theory and Applications |
| Volume | 48 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - Mar 1986 |
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Keywords
- line search
- nonlinear least squares
- sectioning
- Unconstrained optimization
ASJC Scopus subject areas
- Control and Optimization
- Applied Mathematics
- Management Science and Operations Research
Cite this
An efficient line search for nonlinear least squares. / Al-Baali, M.; Fletcher, R.
In: Journal of Optimization Theory and Applications, Vol. 48, No. 3, 03.1986, p. 359-377.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - An efficient line search for nonlinear least squares
AU - Al-Baali, M.
AU - Fletcher, R.
PY - 1986/3
Y1 - 1986/3
N2 - The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength which satisfies certain standard conditions. Prototype algorithms are described which guarantee finding such a step in a finite number of operations. This is achieved by first bracketing an interval of acceptable values and then reducing this bracket uniformly by the repeated use of sectioning in a systematic way. Some new theorems about convergence and termination of the line search are presented. Use of these algorithms to solve the line search subproblem in methods for nonlinear least squares is considered. We show that substantial gains in efficiency can be made by making polynomial interpolations to the individual residual functions rather than the overall objective function. We also study modified schemes in which the Jacobian matrix is evaluated as infrequently as possible, and show that further worthwhile savings can be made. Numerical results are presented.
AB - The line search subproblem in unconstrained optimization is concerned with finding an acceptable steplength which satisfies certain standard conditions. Prototype algorithms are described which guarantee finding such a step in a finite number of operations. This is achieved by first bracketing an interval of acceptable values and then reducing this bracket uniformly by the repeated use of sectioning in a systematic way. Some new theorems about convergence and termination of the line search are presented. Use of these algorithms to solve the line search subproblem in methods for nonlinear least squares is considered. We show that substantial gains in efficiency can be made by making polynomial interpolations to the individual residual functions rather than the overall objective function. We also study modified schemes in which the Jacobian matrix is evaluated as infrequently as possible, and show that further worthwhile savings can be made. Numerical results are presented.
KW - line search
KW - nonlinear least squares
KW - sectioning
KW - Unconstrained optimization
UR - http://www.scopus.com/inward/record.url?scp=0022675909&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0022675909&partnerID=8YFLogxK
U2 - 10.1007/BF00940566
DO - 10.1007/BF00940566
M3 - Article
AN - SCOPUS:0022675909
VL - 48
SP - 359
EP - 377
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
SN - 0022-3239
IS - 3
ER -