Unconstrained Rosenbrock Function
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Unconstrained Rosenbrock Function

The Rosenbrock function is defined as

$\displaystyle f(x) $
$\displaystyle = $
$\displaystyle \frac{1}{2} \{ 100 (x_2 - x_1^2)^2 + (1 - x_1)^2 \} $
$\displaystyle $
$\displaystyle = $
$\displaystyle \frac{1}{2} \{ f_1^2(x) + f_2^2(x) \} , \quad x = (x_1,x_2) $

The minimum function value $f^* = f(x^*) = 0$ is at the point $x^* = (1,1)$.

The following code calls the NLPTR subroutine to solve the optimization problem:

The NLPTR is a trust-region optimization method. The F_ROSEN module represents the Rosenbrock function, and the G_ROSEN module represents its gradient. Specifying the gradient can reduce the number of function calls by the optimization subroutine. The optimization begins at the initial point $x=(-1.2, 1)$. For more information about the NLPTR subroutine and its arguments, see the section NLPTR Call. For details about the options vector, which is given by the OPTN vector in the preceding code, see the section Options Vector.

A portion of the output produced by the NLPTR subroutine is shown in Figure 14.1.

Figure 14.1: NLPTR Solution to the Rosenbrock Problem

Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
1 X1 -1.200000 -107.800000
2 X2 1.000000 -44.000000


Value of Objective Function = 12.1


Trust Region Optimization


Without Parameter Scaling


CRP Jacobian Computed by Finite Differences

Parameter Estimates 2

Optimization Start
Active Constraints 0 Objective Function 12.1
Max Abs Gradient Element 107.8 Radius 1

Iteration   Restarts Function
Calls 		Active
Constraints 		  Objective
Function 		Objective
Function
Change 		Max Abs
Gradient
Element 		Lambda Trust
Region
Radius
1   0 2 0   2.36594 9.7341 2.3189 0 1.000
2   0 5 0   2.05926 0.3067 5.2875 0.385 1.526
3   0 8 0   1.74390 0.3154 5.9934 0 1.086
4   0 9 0   1.43279 0.3111 6.5134 0.918 0.372
5   0 10 0   1.13242 0.3004 4.9245 0 0.373
6   0 11 0   0.86905 0.2634 2.9302 0 0.291
7   0 12 0   0.66711 0.2019 3.6584 0 0.205
8   0 13 0   0.47959 0.1875 1.7354 0 0.208
9   0 14 0   0.36337 0.1162 1.7589 2.916 0.132
10   0 15 0   0.26903 0.0943 3.4089 0 0.270
11   0 16 0   0.16280 0.1062 0.6902 0 0.201
12   0 19 0   0.11590 0.0469 1.1456 0 0.316
13   0 20 0   0.07616 0.0397 0.8462 0.931 0.134
14   0 21 0   0.04873 0.0274 2.8063 0 0.276
15   0 22 0   0.01862 0.0301 0.2290 0 0.232
16   0 25 0   0.01005 0.00858 0.4553 0 0.256
17   0 26 0   0.00414 0.00590 0.4297 0.247 0.104
18   0 27 0   0.00100 0.00314 0.4323 0.0453 0.104
19   0 28 0   0.0000961 0.000906 0.1134 0 0.104
20   0 29 0   1.67875E-6 0.000094 0.0224 0 0.0569
21   0 30 0   6.9583E-10 1.678E-6 0.000336 0 0.0248
22   0 31 0   1.3128E-16 6.96E-10 1.977E-7 0 0.00314

Optimization Results
Iterations 22 Function Calls 32
Hessian Calls 23 Active Constraints 0
Objective Function 1.312819E-16 Max Abs Gradient Element 1.9773301E-7
Lambda 0 Actual Over Pred Change 0
Radius 0.0031402097    

ABSGCONV convergence criterion satisfied.

Optimization Results
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
1 X1 1.000000 0.000000198
2 X2 1.000000 -0.000000105


Value of Objective Function = 1.312819E-16


Since $f(x) = \frac{1}{2} \{ f_1^2(x) + f_2^2(x)\} $, you can also use least squares techniques in this situation. The following code calls the NLPLM subroutine to solve the problem. The output is shown in Figure 14.2.

Figure 14.2: NLPLM Solution Using the Least Squares Technique

Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
1 X1 -1.200000 -107.799999
2 X2 1.000000 -44.000000


Value of Objective Function = 12.1


Levenberg-Marquardt Optimization


Scaling Update of More (1978)


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 107.79999885 Radius 2626.5613241

Iteration   Restarts Function
Calls 		Active
Constraints 		  Objective
Function 		Objective
Function
Change 		Max Abs
Gradient
Element 		Lambda Ratio
Between
Actual
and
Predicted
Change
1   0 4 0   2.18185 9.9181 17.4704 0.00804 0.964
2   0 6 0   1.59370 0.5881 3.7015 0.0190 0.988
3   0 7 0   1.32848 0.2652 7.0843 0.00830 0.678
4   0 8 0   1.03891 0.2896 6.3092 0.00753 0.593
5   0 9 0   0.78943 0.2495 7.2617 0.00634 0.486
6   0 10 0   0.58838 0.2011 7.8837 0.00462 0.393
7   0 11 0   0.34224 0.2461 6.6815 0.00307 0.524
8   0 12 0   0.19630 0.1459 8.3857 0.00147 0.469
9   0 13 0   0.11626 0.0800 9.3086 0.00016 0.409
10   0 14 0   0.0000396 0.1162 0.1781 0 1.000
11   0 15 0   7.5224E-23 0.000040 2.45E-10 0 1.000

Optimization Results
Iterations 11 Function Calls 16
Jacobian Calls 12 Active Constraints 0
Objective Function 7.522424E-23 Max Abs Gradient Element 2.453149E-10
Lambda 0 Actual Over Pred Change 1
Radius 0.0178062366    

ABSGCONV convergence criterion satisfied.

Optimization Results
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
1 X1 1.000000 -2.45315E-10
2 X2 1.000000 1.226574E-10


Value of Objective Function = 7.522424E-23


The Levenberg-Marquardt least squares method, which is the method used by the NLPLM subroutine, is a modification of the trust-region method for nonlinear least squares problems. The F_ROSEN module represents the Rosenbrock function. Note that for least squares problems, the $m$ functions $f_1(x),\ldots , f_ m(x)$ are specified as elements of a vector; this is different from the manner in which $f(x)$ is specified for the other optimization techniques. No derivatives are specified in the preceding code, so the NLPLM subroutine computes finite-difference approximations. For more information about the NLPLM subroutine, see the section NLPLM Call.

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