The Nonlinear Programming Solver
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
![\[ \displaystyle \mathop {\textrm{minimize}}f(x) = \sum _{i=1}^{n-1} ( - 4 x_{i} + 3.0) + \sum _{i=1}^{n-1} (x_{i}^{2} + x_{n}^{2})^{2} \]](images/ormpug_nlpsolver0179.png){kind=small}
and
![\[ \begin{array}{ll} \displaystyle \mathop {\textrm{minimize}}& f(x) = \sum _{i=1}^{n-1} \cos (-.5x_{i+1} - x_ i^2) \\ \textrm{subject to}& 1 \le x_ i \le 2, \; i = 1,\dots , n \end{array} \]](images/ormpug_nlpsolver0180.png){kind=small}
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 10.2.1.
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 / algorithm=activeset;
quit;
The problem and solution summaries are shown in Output 10.2.2.