The Nonlinear Programming Solver

Some NLP problems contain many local minima. By default, the NLP solver converges to a single local minimum. However, the NLP solver can search the feasible region for other local minima. After it completes the search, it returns the point where the objective function achieves its minimum value. (This point might not be a local minimum; see the SOLTYPE= option for more details.) Consider the following example, taken from Hock and Schittkowski (1981):

The following statements call the NLP solver to search the feasible region for different local minima. The PERFORMANCE statement specifies that the multistart algorithm use multiple threads. The SEED= option is specified for reproducibility, but it is not required in running the multistart algorithm.

The PRINT statement prints the initial point that is specified in the VAR declaration (x.init), the starting point that led to the best local solution (x.msinit), and finally the best local solution (x) that is found by the NLP solver in multistart mode.

proc optmodel;
   var x{i in 1..5} >= -5 <= 5 init -2;
   
   min f=(x[1] - 1)^2 + (x[1] - x[2])^2 + (x[2] - x[3])^3 + 
         (x[3] - x[4])^4 + (x[4] - x[5])^4;
         
   con g1: x[1] + x[2]^2 + x[3]^3 =  2 + 3*sqrt(2);
   con g2: x[2] + x[4]   - x[3]^2 = -2 + 2*sqrt(2);
   con g3: x[1]*x[5] = 2;
   
   performance nthreads=4;
   solve with nlp/multistart seed=42 msmaxstarts=10;
   print x.init x.msinit x;
quit;

The SAS log is shown in Output 8.5.1.

The first column in the log indicates the index of the current starting point. An additional indicator (*, r, or R) can appear after the index to provide more information about the optimization run starting from the current point. See the section Iteration Log for Multistart for more information. The second column records the best objective that has been found so far. Columns 3 to 6 report the objective value, infeasibility, optimality error, and number of iterations of the local solver when it was started from the current starting point. Finally, the last column records the status of the local solver—namely, whether it was able to converge to a local optimum from the current starting point.

The summaries and solution are shown in Output 8.5.2. Note that the best solution was found by starting the local solver from a starting point (x.msinit) that is different from the point specified in the VAR declaration (x.init).