The OPTQP Procedure

(View the complete code for this example.)

Consider a small illustrative example. Suppose you want to minimize a two-variable quadratic function on the nonnegative quadrant, subject to two constraints:

The linear objective function coefficients, vector of right-hand sides, and lower and upper bounds are identified immediately as

Carefully construct the quadratic matrix . Observe that you can use symmetry to separate the main-diagonal and off-diagonal elements:

The first expression

sums the main-diagonal elements. Thus, in this case you have

Notice that the main-diagonal values are doubled in order to accommodate the 1/2 factor. Now the second term

sums the off-diagonal elements in the strict lower triangular part of the matrix. The only off-diagonal () term in the objective function is , so you have

Notice that you do not need to specify the upper triangular part of the quadratic matrix.

Finally, the matrix of constraints is as follows:

The SAS input data set with a quadratic programming system (QPS) format for the preceding problem can be expressed in the following manner:

data gsdata;
   input field1 $ field2 $ field3 $ field4 field5 $ field6 @;
   datalines;
NAME     .      EXAMPLE    .             .         .
ROWS     .      .          .             .         .
N        OBJ    .          .             .         .
L        R1     .          .             .         .
G        R2     .          .             .         .
COLUMNS  .      .          .             .         .
.        X1     R1         1.0           R2        1.0
.        X1     OBJ        2.0           .         .
.        X2     R1        -1.0           R2        2.0
.        X2     OBJ        3.0           .         .
RHS      .      .          .             .         .
.        RHS    R1         1.0           .         .
.        RHS    R2         100           .         .
RANGES   .      .          .             .         .
BOUNDS   .      .          .             .         .
QUADOBJ  .      .          .             .         .
.        X1     X1         2.0           .         .
.        X1     X2         2.5           .         .
.        X2     X2         20            .         .
ENDATA   .      .          .             .         .
;

For more details about the QPS-format data set, see Chapter 17, The MPS-Format SAS Data Set.

Alternatively, if you have a QPS-format flat file named gs.qps, then the following call to the SAS macro %MPS2SASD translates that file into a SAS data set, named gsdata:

%mps2sasd(mpsfile =gs.qps, outdata = gsdata);

Note: The SAS macro %MPS2SASD is provided in SAS/OR software. For more information, see Converting an MPS/QPS-Format File: %MPS2SASD.

You can use the following call to PROC OPTQP:

proc optqp data=gsdata
   primalout = gspout
   dualout   = gsdout;
run;

The procedure output is displayed in Figure 2.

The optimal primal solution is displayed in Figure 3.

The SAS log shown in Figure 4 provides information about the problem, convergence information after each iteration, and the optimal objective value.

See the section Interior Point Algorithm: Overview and the section Iteration Log for the OPTQP Procedure for more details about convergence information given by the iteration log.