PROC NLP: MINQUAD and MAXQUAD Statements

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 $\text{[math]}$ , vector $\text{[math]}$ , and scalar $\text{[math]}$ 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 $\text{[math]}$ and $\text{[math]}$ 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 $\text{[math]}$ 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 $\text{[math]}$ and vector $\text{[math]}$ . These element names can be used on the left side of numerical assignments. The one-dimensional index value $\text{[math]}$ following H_name, which corresponds to the element $\text{[math]}$ , is computed by $\text{[math]}$ . The matrix $\text{[math]}$ and vector $\text{[math]}$ are initialized to zero, so that only the nonzero elements must be given. This specification is efficient for small problems with sparse $\text{[math]}$ 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 $\text{[math]}$ and vector $\text{[math]}$ 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;
```