| Language Reference |
NLPHQN Call
calculates hybrid quasi-Newton least squares
- CALL NLPHQN( rc, xr, "fun", x0 <,opt, blc, tc, par, "ptit", "jac">);
See the section "Nonlinear Optimization and Related Subroutines" for a listing of all NLP subroutines. See Chapter 11 for a description of the inputs to and outputs of all NLP subroutines.
The NLPHQN subroutine uses one of the Fletcher and Xu (1987) hybrid quasi-Newton methods. Refer also to Al-Baali and Fletcher (1985, 1986). In each iteration, the subroutine uses a criterion to decide whether a Gauss-Newton or a dual quasi-Newton search direction is appropriate. You can choose one of three criteria (HY1, HY2, or HY3) proposed by Fletcher and Xu (1987) with the sixth element of the opt vector. The default is HY2. The subroutine computes the crossproduct Jacobian (for the Gauss-Newton step), updates the Cholesky factor of an approximate Hessian (for the quasi-Newton step), and performs a line search to compute an approximate minimum along the search direction. The default line-search technique used by the NLPHQN method is designed for least squares problems (refer to Lindström and Wedin 1984, and Al-Baali and Fletcher 1986), but you can specify a different line-search algorithm with the fifth element of the opt argument. See the section "Options Vector" for details.
You can specify two update formulas with the fourth element of the opt argument as indicated in the following table.
| Value of opt[4] | Update Method |
| 1 | Dual Broyden, Fletcher, Goldfarb, and Shanno (DBFGS) update of the Cholesky factor of the Hessian matrix. This is the default. |
| 2 | Dual Davidon, Fletcher, and Powell (DDFP) update of the Cholesky factor of the Hessian matrix. |
The NLPHQN subroutine needs approximately the same amount of working memory as the NLPLM subroutine, and in most applications, the latter seems to be superior. Hence, the NLPHQN method is recommended only when the NLPLM method encounters problems.
Note: In least squares subroutines, you must set the first
element of the opt vector to 
, the number of functions.
In addition to the standard iteration history, the NLPHQN subroutine prints the following information:
- Under the heading Iter, an asterisk (*) printed after the iteration number indicates that, on the basis of the Fletcher and Xu (1987) criterion, the subroutine used a Gauss-Newton search direction instead of a quasi-Newton search direction.
- The heading alpha is the step size,
, computed with the line-search algorithm.
- The heading slope refers to
, the slope of the
search direction at the current parameter iterate 
.
For minimization, this value should
be significantly smaller than zero.
Otherwise, the line-search algorithm has difficulty
reducing the function value sufficiently.
The following statements use the NLPHQN call to solve the unconstrained Rosenbrock problem (see the section "Unconstrained Rosenbrock Function").
title 'Test of NLPHQN subroutine: No Derivatives';
start F_ROSEN(x);
y = j(1,2,0.);
y[1] = 10. * (x[2] - x[1] * x[1]);
y[2] = 1. - x[1];
return(y);
finish F_ROSEN;
x = {-1.2 1.};
optn = {2 2};
call nlphqn(rc,xr,"F_ROSEN",x,optn);
The iteration history for the subroutine follows.
Optimization Start
Parameter Estimates
Gradient
Objective
N Parameter Estimate Function
1 X1 -1.200000 -107.799999
2 X2 1.000000 -44.000000
Value of Objective Function = 12.1
Hybrid Quasi-Newton LS Minimization
Dual Broyden - Fletcher - Goldfarb - Shanno Update (DBFGS)
Version HY2 of Fletcher & Xu (1987)
Gradient Computed by Finite Differences
CRP Jacobian Computed by Finite Differences
Parameter Estimates 2
Functions (Observations) 2
Optimization Start
Active Constraints 0 Objective Function 12.1
Max Abs Gradient Element 107.7999987
Function Active Objective
Iter Restarts Calls Constraints Function
1 0 3 0 7.22423
2* 0 5 0 0.97090
3* 0 7 0 0.81911
4 0 9 0 0.69103
5 0 19 0 0.47345
6* 0 21 0 0.35906
7* 0 22 0 0.23342
8* 0 24 0 0.14799
9* 0 26 0 0.00948
10* 0 28 0 1.98834E-6
11* 0 30 0 7.0768E-10
12* 0 32 0 2.0246E-21
Objective Max Abs Slope of
Function Gradient Step Search
Iter Change Element Size Direction
1 4.8758 56.9322 0.0616 -628.8
2* 6.2533 2.3017 0.266 -14.448
3* 0.1518 3.7839 0.119 -1.942
4 0.1281 5.5103 2.000 -0.144
5 0.2176 8.8638 11.854 -0.194
6* 0.1144 9.8734 0.253 -0.947
7* 0.1256 10.1490 0.398 -0.718
8* 0.0854 11.6248 1.346 -0.467
9* 0.1385 2.6275 1.443 -0.296
10* 0.00947 0.00609 0.938 -0.0190
11* 1.988E-6 0.000748 1.003 -398E-8
12* 7.08E-10 1.82E-10 1.000 -14E-10
Optimization Results
Iterations 12 Function Calls 33
Jacobian Calls 13 Gradient Calls 19
Active Constraints 0 Objective Function 2.024612E-21
Max Abs Gradient Element 1.816863E-10 Slope of Search Direction -1.415366E-9
ABSGCONV convergence criterion satisfied.
Optimization Results
Parameter Estimates
Gradient
Objective
N Parameter Estimate Function
1 X1 1.000000 1.816863E-10
2 X2 1.000000 -1.22069E-10
Value of Objective Function = 2.024612E-21
Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.