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The NLP Procedure
Introductory Examples

The following introductory examples illustrate how to get started using the NLP procedure.

An Unconstrained Problem

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 Newton-Raphson 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, Levenberg-Marquardt). 

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 .

Figure 6.1 Least Squares Minimization
PROC NLP: Least Squares Minimization


Levenberg-Marquardt Optimization


Scaling Update of More (1978)

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   1.9337E-28 7.4115 6.39E-14 0 1.000

Optimization Results
Iterations 2 Function Calls 4
Jacobian Calls 3 Active Constraints 0
Objective Function 1.933695E-28 Max Abs Gradient Element 6.394885E-14
Lambda 0 Actual Over Pred Change 1
Radius 7.7001288198    

ABSGCONV convergence criterion satisfied.

PROC NLP: Least Squares Minimization

Optimization Results
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
1 x1 1.000000 6.394885E-14
2 x2 1.000000 -2.22045E-14


Value of Objective Function = 1.933695E-28

Boundary Constraints on the Decision Variables

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 x1-x2 <= .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.

Figure 6.2 Least Squares with Bounds Solution
PROC NLP: Least Squares Minimization


Scaling Update of More (1978)

PROC NLP: Least Squares Minimization

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  

Linear Constraints on the Decision Variables

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 x1-x2 <= .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.

Figure 6.3 Least Squares with Bounds and Linear Constraints Iteration History
PROC NLP: Least Squares Minimization


Levenberg-Marquardt Optimization


Scaling Update of More (1978)

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 1.233E-9 1.185E-6 0 0.998

Optimization Results
Iterations 5 Function Calls 11
Jacobian Calls 7 Active Constraints 1
Objective Function 0.1665792899 Max Abs Gradient Element 1.1847291E-6
Lambda 0 Actual Over Pred Change 0.9981768536
Radius 0.0000994255    

ABSGCONV convergence criterion satisfied.

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.

Figure 6.4 Least Squares with Bounds and Linear Constraints Solution
PROC NLP: Least Squares Minimization


Scaling Update of More (1978)

PROC NLP: Least Squares Minimization

Optimization Results
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function
1 x1 0.423645 -0.312000
2 x2 0.176355 -0.312000

Linear Constraints Evaluated at Solution
1 ACT 2.7756E-17 = 0.6000 - 1.0000 * x1 - 1.0000 * x2

Nonlinear Constraints on the Decision Variables

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 x1-x2 <= .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.

Figure 6.5 Least Squares with Bounds, Linear and Nonlinear Constraints, Iteration History
PROC NLP: Nonlinear Minimization


Dual Quasi-Newton Optimization


Modified VMCWD Algorithm of Powell (1978, 1982)


Dual Broyden - Fletcher - Goldfarb - Shanno Update (DBFGS)


Lagrange Multiplier Update of Powell(1982)

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 3.92E-6 1.000 0.00200
10   0 18 0.33003 0 2.053E-8 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 9.4437885E-6 Value Lagrange Function 0.3300307155
Maximum Gradient of the Lagran Func 9.1683548E-6 Slope of Search Direction -2.053448E-8

Figure 6.6 Least Squares with Bounds, Linear and Nonlinear Constraints, Solution
PROC NLP: Nonlinear Minimization


Note: At least one element of the (projected) gradient is greater than 1e-3.

PROC NLP: Nonlinear Minimization

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


Value of Objective Function = 0.3300307479

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 2.112E-8 2.112E-8 .    

Not all of the optimization methods support nonlinear constraints. In particular the Levenberg-Marquardt method, the default for LSQ, does not support nonlinear constraints. (For more information about the particular algorithms, see the section Optimization Algorithms.) The Quasi-Newton method is the prime choice for solving nonlinear programs with nonlinear constraints. The option TECH=QUANEW in the PROC NLP statement causes the Quasi-Newton method to be used.

A Simple Maximum Likelihood Example

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 > 1e-12;
   loglik=-0.5*((x-mean)/sigma)**2-log(sigma);
run;

After a few iterations of the default Newton-Raphson optimization algorithm, PROC NLP produces the results shown in Figure 6.7.

Figure 6.7 Maximum Likelihood Estimates
PROC NLP: Nonlinear Maximization

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 -1.33149E-10
2 sigma 2.000000 0.632456 3.162278 0.025031 5.6064146E-9


Value of Objective Function = -5.965735903

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.

Figure 6.8 Hessian Matrix
PROC NLP: Nonlinear Maximization

Hessian Matrix
  mean sigma
mean -1.250000003 1.33149E-10
sigma 1.33149E-10 -2.500000014


Determinant = 3.1250000245


Matrix has Only Negative Eigenvalues

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.

Figure 6.9 Covariance Matrix
PROC NLP: Nonlinear Maximization

Covariance Matrix 2: H = (NOBS/d)
inv(G)
  mean sigma
mean 0.7999999982 4.260769E-11
sigma 4.260769E-11 0.3999999978


Factor sigm = 1


Determinant = 0.3199999975


Matrix has 2 Positive Eigenvalue(s)

The PROFILE statement computes the values of the profile likelihood confidence limits on SIGMA and MEAN, as shown in Figure 6.10.

Figure 6.10 Confidence Limits
PROC NLP: Nonlinear Maximization

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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