The Nonlinear Programming Solver
This example demonstrates the use of the NLP solver to solve the following highly nonlinear optimization problem, which appears in Hock and Schittkowski (1981):
![\[ \begin{array}{ll} \displaystyle \mathop {\textrm{minimize}}& f(x) = 0.4 (x_{1}/x_{7})^{0.67} + 0.4 (x_{2}/x_{8})^{0.67} + 10 - x_{1} - x_{2} \\ \textrm{subject to}& 1 - 0.0588 x_5 x_7 - 0.1 x_1 \ge 0 \\ & 1 - 0.0588x_6 x_8 - 0.1 x_1 - 0.1 x_2 \ge 0 \\ & 1 - 4 x_3/x_5 - 2/(x_3^{0.71} x_5) - 0.0588 x_7/x_3^{1.3} \ge 0 \\ & 1 - 4 x_4/x_6 - 2/(x_4^{0.71} x_6) - 0.0588 x_8/x_4^{1.3} \ge 0 \\ & 0.1 \le f(x) \le 4.2 \\ & 0.1 \le x_ i \le 10, i=1,2, \ldots , 8 \end{array} \]](images/ormpug_nlpsolver0177.png)
The initial point used is 
. You can call the NLP solver within PROC OPTMODEL to solve the problem by writing the following SAS statements:
proc optmodel;
var x{1..8} >= 0.1 <= 10;
min f = 0.4*(x[1]/x[7])^0.67 + 0.4*(x[2]/x[8])^0.67 + 10 - x[1] - x[2];
con c1: 1 - 0.0588*x[5]*x[7] - 0.1*x[1] >= 0;
con c2: 1 - 0.0588*x[6]*x[8] - 0.1*x[1] - 0.1*x[2] >= 0;
con c3: 1 - 4*x[3]/x[5] - 2/(x[3]^0.71*x[5]) - 0.0588*x[7]/x[3]^1.3 >= 0;
con c4: 1 - 4*x[4]/x[6] - 2/(x[4]^0.71*x[6]) - 0.0588*x[8]/x[4]^1.3 >= 0;
con c5: 0.1 <= f <= 4.2;
/* starting point */
x[1] = 6;
x[2] = 3;
x[3] = 0.4;
x[4] = 0.2;
x[5] = 6;
x[6] = 6;
x[7] = 1;
x[8] = 0.5;
solve with nlp / algorithm=activeset;
print x;
quit;
The summaries and the solution are shown in Output 10.1.1.
Output 10.1.1: Summaries and the Returned Solution
The OPTMODEL Procedure
| Problem Summary | |
|---|---|
| Objective Sense | Minimization |
| Objective Function | f |
| Objective Type | Nonlinear |
| Number of Variables | 8 |
| Bounded Above | 0 |
| Bounded Below | 0 |
| Bounded Below and Above | 8 |
| Free | 0 |
| Fixed | 0 |
| Number of Constraints | 5 |
| Linear LE (<=) | 0 |
| Linear EQ (=) | 0 |
| Linear GE (>=) | 0 |
| Linear Range | 0 |
| Nonlinear LE (<=) | 0 |
| Nonlinear EQ (=) | 0 |
| Nonlinear GE (>=) | 4 |
| Nonlinear Range | 1 |