The NLP Procedure 
Introductory Examples 
The following introductory examples illustrate how to get started using the NLP procedure.
Consider the simple example of minimizing the Rosenbrock function (Rosenbrock 1960):
The minimum function value is at . This problem does not have any constraints.
The following statements can be used to solve this problem:
proc nlp; min f; decvar x1 x2; f1 = 10 * (x2  x1 * x1); f2 = 1  x1; f = .5 * (f1 * f1 + f2 * f2); run;
The MIN statement identifies the symbol f that characterizes the objective function in terms of f1 and f2, and the DECVAR statement names the decision variables x1 and x2. Because there is no explicit optimizing algorithm option specified (TECH=), PROC NLP uses the NewtonRaphson method with ridging, the default algorithm when there are no constraints.
A better way to solve this problem is to take advantage of the fact that is a sum of squares of and and to treat it as a least squares problem. Using the LSQ statement instead of the MIN statement tells the procedure that this is a least squares problem, which results in the use of one of the specialized algorithms for solving least squares problems (for example, LevenbergMarquardt).
proc nlp; lsq f1 f2; decvar x1 x2; f1 = 10 * (x2  x1 * x1); f2 = 1  x1; run;
The LSQ statement results in the minimization of a function that is the sum of squares of functions that appear in the LSQ statement. The least squares specification is preferred because it enables the procedure to exploit the structure in the problem for numerical stability and performance.
PROC NLP displays the iteration history and the solution to this least squares problem as shown in Figure 6.1. It shows that the solution has and . As expected in an unconstrained problem, the gradient at the solution is very close to .
Parameter Estimates  2 

Functions (Observations)  2 
Optimization Start  

Active Constraints  0  Objective Function  7.7115046337 
Max Abs Gradient Element  38.778863865  Radius  450.91265904 
Iteration  Restarts  Function Calls 
Active Constraints 
Objective Function 
Objective Function Change 
Max Abs Gradient Element 
Lambda  Ratio Between Actual and Predicted Change 


1  0  2  0  7.41150  0.3000  77.0013  0  0.0389  
2  0  3  0  1.9337E28  7.4115  6.39E14  0  1.000 
Optimization Results  

Iterations  2  Function Calls  4 
Jacobian Calls  3  Active Constraints  0 
Objective Function  1.933695E28  Max Abs Gradient Element  6.394885E14 
Lambda  0  Actual Over Pred Change  1 
Radius  7.7001288198 
Bounds on the decision variables can be used. Suppose, for example, that it is necessary to constrain the decision variables in the previous example to be less than . That can be done by adding a BOUNDS statement.
proc nlp; lsq f1 f2; decvar x1 x2; bounds x1x2 <= .5; f1 = 10 * (x2  x1 * x1); f2 = 1  x1; run;
The solution in Figure 6.2 shows that the decision variables meet the constraint bounds.
Optimization Results  

Parameter Estimates  
N  Parameter  Estimate  Gradient Objective Function 
Active Bound Constraint 
1  x1  0.500000  0.500000  Upper BC 
2  x2  0.250000  0 
More general linear equality or inequality constraints of the form
can be specified in a LINCON statement. For example, suppose that in addition to the bounds constraints on the decision variables it is necessary to guarantee that the sum is less than or equal to . That can be achieved by adding a LINCON statement:
proc nlp; lsq f1 f2; decvar x1 x2; bounds x1x2 <= .5; lincon x1 + x2 <= .6; f1 = 10 * (x2  x1 * x1); f2 = 1  x1; run;
The output in Figure 6.3 displays the iteration history and the convergence criterion.
Parameter Estimates  2 

Functions (Observations)  2 
Lower Bounds  0 
Upper Bounds  2 
Linear Constraints  1 
Iteration  Restarts  Function Calls 
Active Constraints 
Objective Function 
Objective Function Change 
Max Abs Gradient Element 
Lambda  Ratio Between Actual and Predicted Change 


1  0  2  0  '  0.23358  3.6205  3.3399  0  0.939  
2  1  6  0  '  0.16687  0.0667  0.4865  174.8  0.535  
3  2  8  1  0.16679  0.000084  0.2677  0.00430  0.0008  
4  2  9  1  0.16658  0.000209  0.000650  0  0.998  
5  2  10  1  0.16658  1.233E9  1.185E6  0  0.998 
Optimization Results  

Iterations  5  Function Calls  11 
Jacobian Calls  7  Active Constraints  1 
Objective Function  0.1665792899  Max Abs Gradient Element  1.1847291E6 
Lambda  0  Actual Over Pred Change  0.9981768536 
Radius  0.0000994255 
Figure 6.4 shows that the solution satisfies the linear constraint. Note that the procedure displays the active constraints (the constraints that are tight) at optimality.
Optimization Results  

Parameter Estimates  
N  Parameter  Estimate  Gradient Objective Function 
1  x1  0.423645  0.312000 
2  x2  0.176355  0.312000 
More general nonlinear equality or inequality constraints can be specified using an NLINCON statement. Consider the least squares problem with the additional constraint
This constraint is specified by a new function c1 constrained to be greater than or equal to 0 in the NLINCON statement. The function c1 is defined in the programming statements.
proc nlp tech=QUANEW; min f; decvar x1 x2; bounds x1x2 <= .5; lincon x1 + x2 <= .6; nlincon c1 >= 0; c1 = x1 * x1  2 * x2; f1 = 10 * (x2  x1 * x1); f2 = 1  x1; f = .5 * (f1 * f1 + f2 * f2); run;
Figure 6.5 shows the iteration history, and Figure 6.6 shows the solution to this problem.
Parameter Estimates  2 

Lower Bounds  0 
Upper Bounds  2 
Linear Constraints  1 
Nonlinear Constraints  1 
Optimization Start  

Objective Function  3.6940664349  Maximum Constraint Violation  0 
Maximum Gradient of the Lagran Func  24.167449944 
Iteration  Restarts  Function Calls 
Objective Function 
Maximum Constraint Violation 
Predicted Function Reduction 
Step Size 
Maximum Gradient Element of the Lagrange Function 


1  0  9  1.33999  0  1.1315  0.558  7.172  
2  0  10  0.81134  0  0.2944  1.000  2.896  
3  0  11  0.61022  0  0.1518  1.000  2.531  
4  0  12  0.49146  0  0.1575  1.000  1.736  
5  0  13  0.37940  0  0.0957  1.000  0.464  
6  0  14  0.34677  0  0.0367  1.000  0.603  
7  0  15  0.33136  0  0.00254  1.000  0.257  
8  0  16  0.33020  0  0.000332  1.000  0.0218  
9  0  17  0.33003  0  3.92E6  1.000  0.00200  
10  0  18  0.33003  0  2.053E8  1.000  0.00002 
Optimization Results  

Iterations  10  Function Calls  19 
Gradient Calls  13  Active Constraints  1 
Objective Function  0.3300307258  Maximum Constraint Violation  0 
Maximum Projected Gradient  9.4437885E6  Value Lagrange Function  0.3300307155 
Maximum Gradient of the Lagran Func  9.1683548E6  Slope of Search Direction  2.053448E8 
Note:  At least one element of the (projected) gradient is greater than 1e3. 
Optimization Results  

Parameter Estimates  
N  Parameter  Estimate  Gradient Objective Function 
Gradient Lagrange Function 
1  x1  0.246960  0.753147  0.753147 
2  x2  0.030495  3.049459  3.049459 
Linear Constraints Evaluated at Solution  

1  0.32255  =  0.6000    1.0000  *  x1    1.0000  *  x2 
Values of Nonlinear Constraints  

Constraint  Value  Residual  Lagrange Multiplier 

[  2  ]  c1_G  2.112E8  2.112E8  . 
Not all of the optimization methods support nonlinear constraints. In particular the LevenbergMarquardt method, the default for LSQ, does not support nonlinear constraints. (For more information about the particular algorithms, see the section Optimization Algorithms.) The QuasiNewton method is the prime choice for solving nonlinear programs with nonlinear constraints. The option TECH=QUANEW in the PROC NLP statement causes the QuasiNewton method to be used.
The following is a very simple example of a maximum likelihood estimation problem with the log likelihood function:
The maximum likelihood estimates of the parameters and form the solution to
where
In the following DATA step, values for are input into SAS data set X; this data set provides the values of .
data x; input x @@; datalines; 1 3 4 5 7 ;
In the following statements, the DATA=X specification drives the building of the objective function. When each observation in the DATA=X data set is read, a new term using the value of is added to the objective function LOGLIK specified in the MAX statement.
proc nlp data=x vardef=n covariance=h pcov phes; profile mean sigma / alpha=.5 .1 .05 .01; max loglik; parms mean=0, sigma=1; bounds sigma > 1e12; loglik=0.5*((xmean)/sigma)**2log(sigma); run;
After a few iterations of the default NewtonRaphson optimization algorithm, PROC NLP produces the results shown in Figure 6.7.
Optimization Results  

Parameter Estimates  
N  Parameter  Estimate  Approx Std Err 
t Value  Approx Pr > t 
Gradient Objective Function 
1  mean  4.000000  0.894427  4.472136  0.006566  1.33149E10 
2  sigma  2.000000  0.632456  3.162278  0.025031  5.6064146E9 
In unconstrained maximization, the gradient (that is, the vector of first derivatives) at the solution must be very close to zero and the Hessian matrix at the solution (that is, the matrix of second derivatives) must have nonpositive eigenvalues. The Hessian matrix is displayed in Figure 6.8.
Hessian Matrix  

mean  sigma  
mean  1.250000003  1.33149E10 
sigma  1.33149E10  2.500000014 
Under reasonable assumptions, the approximate standard errors of the estimates are the square roots of the diagonal elements of the covariance matrix of the parameter estimates, which (because of the COV=H specification) is the same as the inverse of the Hessian matrix. The covariance matrix is shown in Figure 6.9.
Covariance Matrix 2: H = (NOBS/d) inv(G) 


mean  sigma  
mean  0.7999999982  4.260769E11 
sigma  4.260769E11  0.3999999978 
The PROFILE statement computes the values of the profile likelihood confidence limits on SIGMA and MEAN, as shown in Figure 6.10.
Wald and PL Confidence Limits  

N  Parameter  Estimate  Alpha  Profile Likelihood Confidence Limits 
Wald Confidence Limits  
1  mean  4.000000  0.500000  3.384431  4.615569  3.396718  4.603282 
1  mean  .  0.100000  2.305716  5.694284  2.528798  5.471202 
1  mean  .  0.050000  1.849538  6.150462  2.246955  5.753045 
1  mean  .  0.010000  0.670351  7.329649  1.696108  6.303892 
2  sigma  2.000000  0.500000  1.638972  2.516078  1.573415  2.426585 
2  sigma  .  0.100000  1.283506  3.748633  0.959703  3.040297 
2  sigma  .  0.050000  1.195936  4.358321  0.760410  3.239590 
2  sigma  .  0.010000  1.052584  6.064107  0.370903  3.629097 
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