The Nonlinear Programming Solver

Although the NLP solver does not implement techniques that are specialized for least squares problems, this example illustrates how the NLP solver can use general nonlinear optimization techniques to solve least squares problems.

The Bard function, , (Moré, Garbow, and Hillstrom 1981) is a least squares problem that has parameters and functions ,

where the functions are defined as

with , , , and

Starting from , you can reach the minimum at the point , with corresponding objective value = 4.107E–3. You can use the following SAS code to formulate and solve this problem:

proc optmodel;
   set S = 1..15;
   number u{i in S} = i;
   number v{i in S} = 16 - i;
   number w{i in S} = min(u[i], v[i]);
   number y{S} = [ .14 .18 .22 .25 .29 .32 .35 .39 .37 .58
                   .73 .96 1.34 2.10 4.39 ];
   var x{1..3} init 1;
   min f = 0.5 * sum{i in S} ( y[i] -
             ( x[1] + u[i]/(v[i] * x[2] + w[i] * x[3]) )
           )^2;
   solve with nlp / algorithm=interiorpoint;
   print x;
quit;

The output that summarizes the problem characteristics and the solution that the INTERIORPOINT solver obtains are displayed in Figure 3.

The SAS log shown in Figure 4 displays a brief summary of the problem being solved, followed by the iterations that are generated by the solver.