The Unconstrained Nonlinear Programming Solver: Example 11.2: Solving the Accumulated Rosenbrock Function :: SAS/OR(R) 9.2 User's Guide: Mathematical Programming

The Unconstrained Nonlinear Programming Solver

Example 11.2: Solving the Accumulated Rosenbrock Function

Suppose you want to solve the accumulated Rosenbrock function comprising 1000 variables. For ease of understanding the formulation, define ${\cal i}$ to be an ordered set of indices of the decision variables and ${\cal j}$ to be an ordered set of all indices in ${\cal i}$ except the last one. In other words,

${\cal i} & \equiv & \{1,2, ... ,1000\} \ {\cal j} & \equiv & \{1, 2, ... , 999 \}$

The mathematical formulation of the problem is as follows:

$\min \sum_{ j \in {\cal j}} (1 - x_j)^2 + 100(x_{j+1} - x_j^2)^2$

You can use the following SAS code to formulate and solve the problem by using PROC OPTMODEL:

    proc optmodel;
 
      set setI = {1 .. 1000};         /* declare index sets */
      set setJ = setI diff {1000};
      set elementsToPrint = {1 .. 10};
 
      var x {setI};
 
      minimize z = sum{j in setJ} ((1 - x[j])^2 +
                  100*(x[j + 1] - x[j]^2)^2);
 
      solve with nlpu / printfreq = 0;
      print {k in elementsToPrint} x[k];
 
    quit;

The Rosenbrock function has a unique global minimizer . The optimal solution is displayed in Output 11.2.1.

Output 11.2.1: Optimal Solution to the Accumulated Rosenbrock Function

The OPTMODEL Procedure

Problem Summary
Objective Sense	Minimization
Objective Function	z
Objective Type	Nonlinear

Number of Variables	1000
Bounded Above	0
Bounded Below	0
Bounded Below and Above	0
Free	1000
Fixed	0

Number of Constraints	0

The OPTMODEL Procedure

Solution Summary
Solver	L-BFGS
Objective Function	z
Solution Status	Optimal
Objective Value	5.140353E-11
Iterations	44

Optimality Error	5.2161487E-6

[1]
1	1
2	1
3	1
4	1
5	1
6	1
7	1
8	1
9	1
10	1

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