The Nonlinear Programming Solver

Example 7.2 Solving Unconstrained and Bound-Constrained Optimization Problems

Although the NLP techniques are suited for solving generally constrained nonlinear optimization problems, these techniques can also be used to solve unconstrained and bound-constrained problems efficiently. This example considers the relatively large nonlinear optimization problems

and

with . These problems are unconstrained and bound-constrained, respectively.

For large-scale problems, the default memory limit might be too small, which can lead to out-of-memory status. To prevent this occurrence, it is recommended that you set a larger memory size. See the section Memory Limit for more information.

To solve the first problem, you can write the following statements:

proc optmodel;
   number N=100000;
   var x{1..N} init 1.0;
   
   minimize f = sum {i in 1..N - 1} (-4 * x[i] + 3.0)  + 
                sum {i in 1..N - 1} (x[i]^2 + x[N]^2)^2;

   solve with nlp;
quit;

The problem and solution summaries are shown in Output 7.2.1.

To solve the second problem, you can write the following statements (here the active-set method is specifically selected):

proc optmodel;
   number N=100000;
   var x{1..N} >= 1 <= 2;
   
   minimize f = sum {i in 1..N - 1} cos(-0.5*x[i+1] - x[i]^2);

   solve with nlp / tech=ActiveSet;
quit;

The problem and solution summaries are shown in Output 7.2.2.