An Unconstrained Problem

Consider the following example of minimizing the Rosenbrock function (Rosenbrock; 1960):

where . The minimum function value is at . Note that this problem has no constraints.

The following PROC OPTMODEL statements can be used to solve this problem:

proc optmodel;
   number a = 100;
   var x{1..2};
   min f = a*(x[2] - x[1]^2)^2 + (1 - x[1])^2;
   solve with nlpc / tech=newtyp;
   print x;
quit;

The VAR statement declares the decision variables and . The MIN statement identifies the symbol that defines the objective function in terms of and . The TECH=NEWTYP option in the SOLVE statement specifies that the Newton-type method with line search is used to solve this problem. Finally, the PRINT statement is specified to display the solution to this problem.

The output that summarizes the problem characteristics and the solution obtained by the solver are displayed in Figure 12.1. Note that the solution has and , and an objective value very close to .

Bound Constraints on the Decision Variables

Decision variables often have bound constraints of the form

where is the number of decision variables. The bounds on the variables can be specified with the symbols ">=" and "<=" in the VAR statement.

Consider the following bound constrained problem (Hock and Schittkowski; 1981, Example 5):

Given a starting point at , there is a local minimum at

with the objective value . This problem can be formulated and solved by the following statements:

proc optmodel;
   set S = 1..2;
   number lb{S} = [-1.5 -3];
   number ub{S} = [4 3];
   number x0{S} = [0 0];
   var x{i in S} >= lb[i] <= ub[i] init x0[i];
   
   min obj = sin(x[1] + x[2]) + (x[1] - x[2])^2
               - 1.5*x[1] + 2.5*x[2] + 1;
   solve with nlpc / printfreq=1;
   print x;
quit;

The starting point is specified with the keyword INIT in the VAR statement. As usual, the MIN statement identifies the objective function. Since there is no explicit optimization technique specified (using the TECH= option), the NLPC solver uses the trust region method, which is the default algorithm for problems of this size. The PRINTFREQ= option is used to display the iteration log during the optimization process.

In Figure 12.2, the problem is summarized at the top. Then, the details of the iterations are displayed. A message is printed to indicate that the default optimality criteria (ABSOPTTOL=, OPTTOL=E) were satisfied. (See the section Optimality Control for more information.) A summary of the solution shows that the trust region method was used for the optimization and the solution found is optimal. It also shows the optimal objective value and the number of iterations taken to find the solution. The optimal solution is displayed at the end.