The OPTQP Procedure

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 input data table 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 information about the QPS-format data table, see Chapter 8, The MPS-Format Data Table.

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 optimization software. For more information, see the section Converting an MPS/QPS-Format File: %MPS2SASD.

To make the created data set available in the CAS server, you need to upload it as in the following statements. At the same time, you also need to add an _id_ column to ensure that the MPS reader can read the file in the correct order even if it is distributed across several machines and hence loses its original ordering. For more information about the MPS format, see Chapter 8, The MPS-Format Data Table.

data mylib.gsdata;
   set gsdata;
   _id_ = _n_;
run;

Alternatively, you can create the MPS data table directly in CAS and include the _id_ column in the datalines input.

You can use the following call to PROC OPTQP:

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

The procedure output is displayed in Figure 2.

The optimal primal solution is stored in the data table gspout and is displayed in Figure 3.

proc print data = mylib.gspout label;
run;

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 more details about convergence information given by the iteration log.