The Unconstrained Nonlinear Programming Solver: Solving a Highly Nonlinear Problem :: SAS/OR(R) 9.22 User's Guide: Mathematical Programming

The Unconstrained Nonlinear Programming Solver

Example 13.1 Solving a Highly Nonlinear Problem

Consider the following example of minimizing a nonlinear function of three variables, $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ :

$\text{[math]}$

You can use the following SAS statements to formulate and solve the problem in PROC OPTMODEL:


proc optmodel;
  var x, y, z; 
  minimize obj = x^2 + exp(y/10 + 10) + sin(z*y);
  solve with nlpu / tech = fletreev  maxiter   = 100  
                    opttol = 1e-7;
  print x y z;
quit;

The optimal solution is displayed in Output 13.1.1.

Output 13.1.1 Optimal Solution

The OPTMODEL Procedure

Problem Summary
Objective Sense	Minimization
Objective Function	obj
Objective Type	Nonlinear

Number of Variables	3
Bounded Above	0
Bounded Below	0
Bounded Below and Above	0
Free	3
Fixed	0

Number of Constraints	0

Solution Summary
Solver	NLPU/Fletcher-Reeves
Objective Function	obj
Solution Status	Optimal
Objective Value	-1
Iterations	7

Optimality Error	2.714258E-10

x	y	z
0	-2943.7	0.090182

The following iteration log displays information about the type of algorithm used, objective value, relative gradient norm, and number of function evaluations at each iteration.

Output 13.1.2 Iteration Log

line

NOTE: The problem has 3 variables (3 free, 0 fixed).
NOTE: The problem has 0 linear constraints (0 LE, 0 EQ, 0 GE, 0 range).
NOTE: The problem has 0 nonlinear constraints (0 LE, 0 EQ, 0 GE, 0 range).
NOTE: The OPTMODEL presolver removed 0 variables, 0 linear constraints, and 0
nonlinear constraints.
NOTE: Using analytic derivatives for objective.
NOTE: The Fletcher-Reeves Conjugate Gradient algorithm for unconstrained
optimization is called.
Objective Optimality Function
Iter Value Error Calls
0 22026.5 2202.64657948 1
1 3.2096359E-124 0.99999994 5
2 -0.79979862 0.60026837 16
3 -0.99009673 0.14038682 30
4 -0.99973800 0.02288966 41
5 -0.99999990 0.00045001 48
6 -1.00000000 0.00004279639 52
7 -1.00000000 2.71425843E-10 55
NOTE: Optimal.
NOTE: Objective = -1.
NOTE: At least one W.D format was too small for the number to be printed. The
decimal may be shifted by the "BEST" format.

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