The Rosenbrock Problem

You can use parameters to produce a clear formulation of a problem. Consider the Rosenbrock problem,

where is a parameter (constant), and are optimization variables (whose values are to be determined), and is an objective function.

Here is a PROC OPTMODEL program that solves the Rosenbrock problem:

```proc optmodel;
number alpha = 100; /* declare parameter */
var x {1..2};       /* declare variables */
/* objective function */
min f = alpha*(x[2] - x[1]**2)**2 +
(1 - x[1])**2;
/* now run the solver */
solve;

print x;
quit;
```

The PROC OPTMODEL output is shown in Figure 4.3.

Figure 4.3 Rosenbrock Function Results
The OPTMODEL Procedure

Problem Summary
Objective Sense Minimization
Objective Function f
Objective Type Nonlinear

Number of Variables 2
Bounded Above 0
Bounded Below 0
Bounded Below and Above 0
Free 2
Fixed 0

Number of Constraints 0

Solution Summary
Solver NLP/INTERIORPOINT
Objective Function f
Solution Status Optimal
Objective Value 8.206033E-23
Iterations 14

Optimality Error 9.707102E-11
Infeasibility 0

[1] x
1 1
2 1