The NLP Procedure

MINQUAD and MAXQUAD Statements

MINQUAD H_name [ , g_name [ , c_number ] ] ;

MAXQUAD H_name [ , g_name [ , c_number ] ] ;

The MINQUAD and MAXQUAD statements specify the matrix $H$ , vector $g$ , and scalar $c$ that define a quadratic objective function. The MINQUAD statement is for minimizing the objective function and the MAXQUAD statement is for maximizing the objective function.

The rows and columns in $H$ and $g$ correspond to the order of decision variables given in the DECVAR statement. Specifying the objective function with a MINQUAD or MAXQUAD statement indirectly defines the analytic derivatives for the objective function. Therefore, statements specifying derivatives are not valid in these cases. Also, only use these statements when TECH=LICOMP or TECH=QUADAS and no nonlinear constraints are imposed.

There are three ways of using the MINQUAD or MAXQUAD statement:

Using ARRAY Statements: The names H_name and g_name specified in the MINQUAD or MAXQUAD statement can be used in ARRAY statements. This specification is mainly for small problems with almost dense $H$ matrices.
```
proc nlp pall;
   array h[2,2] .4 0
                 0 4;
   minquad h, -100;
   decvar x1 x2 = -1;
   bounds  2 <= x1 <= 50,
         -50 <= x2 <= 50;
   lincon 10 <= 10 * x1 - x2;
run;
```
Using Elementwise Setting: The names H_name and g_name specified in the MINQUAD or MAXQUAD statement can be followed directly by one-dimensional indices specifying the corresponding elements of the matrix $H$ and vector $g$ . These element names can be used on the left side of numerical assignments. The one-dimensional index value $l$ following H_name, which corresponds to the element $H_{ij}$ , is computed by $l = (i-1)n + j, i \geq j$ . The matrix $H$ and vector $g$ are initialized to zero, so that only the nonzero elements must be given. This specification is efficient for small problems with sparse $H$ matrices.
```
proc nlp pall;
   minquad h, -100;
   decvar x1 x2;
   bounds  2 <= x1 <= 50,
         -50 <= x2 <= 50;
   lincon 10 <= 10 * x1 - x2;
   h1 = .4; h4 = 4;
run;
```
Using MATRIX Statements: The names H_name and g_name specified in the MINQUAD or MAXQUAD statement can be used in MATRIX statements. There are different ways to specify the nonzero elements of the matrix $H$ and vector $g$ by MATRIX statements. The following example illustrates one way to use the MATRIX statement.
```
proc nlp all;
   matrix h[1,1] = .4 4;
   minquad h, -100;
   decvar x1 x2 = -1;
   bounds 2 <= x1 <= 50,
        -50 <= x2 <= 50;
   lincon 10 <= 10 * x1 - x2;
run;
```