The Rosenbrock Problem :: SAS/OR(R) 12.3 User's Guide: Mathematical Programming

The Rosenbrock Problem

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

$\displaystyle \mathop \textrm{minimize}\; f(x_1, x_2) = \alpha \, (x_2 - x_1^2)^2 + (1 - x_1)^2$

where $\alpha = 100$ is a parameter (constant), $x_1$ and $x_2$ are optimization variables (whose values are to be determined), and $f(x_1, x_2)$ 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 5.3.

Figure 5.3: 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

Performance Information
Execution Mode	Single-Machine
Number of Threads	4

Solution Summary
Solver	NLP
Algorithm	Interior Point
Objective Function	f
Solution Status	Optimal
Objective Value	8.206033E-23

Optimality Error	9.707102E-11
Infeasibility	0

Iterations	14
Presolve Time	0.00
Solution Time	0.01

[1]	x
1	1
2	1