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Nonlinear Optimization Examples

Getting Started

Unconstrained Rosenbrock Function

The Rosenbrock function is defined as

     
     

The minimum function value is at the point .

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 . 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 , 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 functions are specified as elements of a vector; this is different from the manner in which 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.

Constrained Betts Function

The linearly constrained Betts function (Hock and Schittkowski; 1981) is defined as

     

The boundary constraints are

     
     

The linear constraint is

     

The following code calls the NLPCG subroutine to solve the optimization problem. The infeasible initial point is specified, and a portion of the output is shown in Figure 14.3.

The NLPCG subroutine performs conjugate gradient optimization. It requires only function and gradient calls. The F_BETTS module represents the Betts function, and since no module is defined to specify the gradient, first-order derivatives are computed by finite-difference approximations. For more information about the NLPCG subroutine, see the section NLPCG Call. For details about the constraint matrix, which is represented by the CON matrix in the preceding code, see the section Parameter Constraints.

Figure 14.3 NLPCG Solution to Betts Problem
Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
Lower
Bound
Constraint
Upper
Bound
Constraint
1 X1 6.800000 0.136000 2.000000 50.000000
2 X2 -1.000000 -2.000000 -50.000000 50.000000

Linear Constraints
1 59.00000 :   10.0000 <= + 10.0000 * X1 - 1.0000 * X2

Parameter Estimates 2
Lower Bounds 2
Upper Bounds 2
Linear Constraints 1

Optimization Start
Active Constraints 0 Objective Function -98.5376
Max Abs Gradient Element 2    

Iteration   Restarts Function
Calls
Active
Constraints
  Objective
Function
Objective
Function
Change
Max Abs
Gradient
Element
Step
Size
Slope of
Search
Direction
1   0 3 0   -99.54682 1.0092 0.1346 0.502 -4.018
2   1 7 1   -99.96000 0.4132 0.00272 34.985 -0.0182
3   2 9 1   -99.96000 1.851E-6 0 0.500 -74E-7

Optimization Results
Iterations 3 Function Calls 10
Gradient Calls 9 Active Constraints 1
Objective Function -99.96 Max Abs Gradient Element 0
Slope of Search Direction -7.403546E-6    

Optimization Results
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
Active
Bound
Constraint
1 X1 2.000000 0.040000 Lower BC
2 X2 5.627884E-13 0  

Linear Constraints Evaluated at Solution
1   10.00000 = -10.0000 + 10.0000 * X1 - 1.0000 * X2

Since the initial point is infeasible, the subroutine first computes a feasible starting point. Convergence is achieved after three iterations, and the optimal point is given to be with an optimal function value of . For more information about the printed output, see the section Printing the Optimization History.

Rosen-Suzuki Problem

The Rosen-Suzuki problem is a function of four variables with three nonlinear constraints on the variables. It is taken from problem 43 of Hock and Schittkowski (1981). The objective function is

     

The nonlinear constraints are

     
     
     

Since this problem has nonlinear constraints, only the NLPQN and NLPNMS subroutines are available to perform the optimization. The following code solves the problem with the NLPQN subroutine:

The F_HS43 module specifies the objective function, and the C_HS43 module specifies the nonlinear constraints. The OPTN vector is passed to the subroutine as the OPT input argument. See the section Options Vector for more information. The value of OPTN[10] represents the total number of nonlinear constraints, and the value of OPTN[11] represents the number of equality constraints. In the preceding code, OPTN[10]=3 and OPTN[11]=0, which indicate that there are three constraints, all of which are inequality constraints. In the subroutine calls, instead of separating missing input arguments with commas, you can specify optional arguments with keywords, as in the CALL NLPQN statement in the preceding code. For details about the CALL NLPQN statement, see the section NLPQN Call.

The initial point for the optimization procedure is , and the optimal point is , with an optimal function value of . Part of the output produced is shown in Figure 14.4.

Figure 14.4 Solution to the Rosen-Suzuki Problem by the NLPQN Subroutine
Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
Gradient
Lagrange
Function
1 X1 1.000000 -3.000000 -3.000000
2 X2 1.000000 -3.000000 -3.000000
3 X3 1.000000 -17.000000 -17.000000
4 X4 1.000000 9.000000 9.000000

Parameter Estimates 4
Nonlinear Constraints 3

Optimization Start
Objective Function -19 Maximum Constraint Violation 0
Maximum Gradient of the Lagran Func 17    

Iteration   Restarts Function
Calls
Objective
Function
Maximum
Constraint
Violation
Predicted
Function
Reduction
Step
Size
Maximum
Gradient
Element
of the
Lagrange
Function
1   0 2 -41.88007 1.8988 13.6803 1.000 5.647
2   0 3 -48.83264 3.0280 9.5464 1.000 5.041
3   0 4 -45.33515 0.5452 2.6179 1.000 1.061
4   0 5 -44.08667 0.0427 0.1732 1.000 0.0297
5   0 6 -44.00011 0.000099 0.000218 1.000 0.00906
6   0 7 -44.00001 2.573E-6 0.000014 1.000 0.00219
7   0 8 -44.00000 9.118E-8 5.098E-7 1.000 0.00022

Optimization Results
Iterations 7 Function Calls 9
Gradient Calls 9 Active Constraints 2
Objective Function -44.00000026 Maximum Constraint Violation 9.1181918E-8
Maximum Projected Gradient 0.0002260958 Value Lagrange Function -44
Maximum Gradient of the Lagran Func 0.0002211456 Slope of Search Direction -5.098303E-7

Optimization Results
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
Gradient
Lagrange
Function
1 X1 -0.000001247 -5.000003 -0.000012904
2 X2 1.000027 -2.999945 0.000221
3 X3 1.999993 -13.000027 -0.000053971
4 X4 -1.000003 4.999994 -0.000020534

In addition to the standard iteration history, the NLPQN subroutine includes the following information for problems with nonlinear constraints:

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