The NLP Procedure

Example 7.2 Using the INQUAD= Option

This example illustrates the INQUAD= option for specifying a quadratic programming problem:

$\min \quad f(x) = \frac{1}{2} x^ T G x + g^ T x + c, \quad \mbox{with} \quad G^ T = G$

Suppose that $c=-100$ , $G=\mr {diag}(.4,4)$ and $2 \leq x_1 \leq 50$ , $-50 \leq x_2 \leq 50$ , and $10 \leq 10x_1 - x_2$ .

You specify the constant $c$ and the Hessian $G$ in the data set QUAD1. Notice that the _TYPE_ variable contains the keywords that identify how the procedure should interpret the observations.

data quad1;
   input _type_ $ _name_ $ x1 x2;
   datalines;
const   .   -100  -100
quad   x1    0.4     0
quad   x2      0     4
;

You specify the QUAD1 data set with the INQUAD= option. Notice that the names of the variables in the QUAD1 data set and the _NAME_ variable match the names of the parameters in the PARMS statement.

proc nlp inquad=quad1 all;
   min ;
   parms x1 x2 = -1;
   bounds  2 <= x1 <= 50,
         -50 <= x2 <= 50;
   lincon 10 <= 10 * x1 - x2;
run;

Alternatively, you can use a sparse format for specifying the $G$ matrix, eliminating the zeros. You use the special variables _ROW_, _COL_, and _VALUE_ to give the nonzero row and column names and value.

data quad2;
   input _type_ $ _row_ $ _col_ $ _value_;
   datalines;
const  .      .    -100
quad   x1    x1     0.4
quad   x2    x2       4
;

You can also include the constraints in the QUAD data set. Notice how the _TYPE_ variable contains keywords that identify how the procedure is to interpret the values in each observation.

data quad3;
   input _type_ $ _name_ $ x1   x2   _rhs_;
   datalines;
const      .     -100  -100   .
quad       x1    0.02     0   .
quad       x2    0.00     2   .
parms      .       -1    -1   .
lowerbd    .        2   -50   .
upperbd    .       50    50   .
ge         .       10    -1  10
;

proc nlp inquad=quad3;
   min ;
   parms x1 x2;
run;