The OPTQP Procedure

The linear least squares problem arises in the context of determining a solution to an overdetermined set of linear equations. In practice, these equations could arise in data fitting and estimation problems. An overdetermined system of linear equations can be defined as

where , , , and . Since this system usually does not have a solution, you need to be satisfied with some sort of approximate solution. The most widely used approximation is the least squares solution, which minimizes .

This problem is called a least squares problem for the following reason. Let , , and be defined as previously. Let be the kth component of the vector :

By definition of the Euclidean norm, the objective function can be expressed as follows:

Therefore, the function you minimize is the sum of squares of m terms ; hence the term least squares. The following example is an illustration of the linear least squares problem; that is, each of the terms is a linear function of x.function plus a constant, .

Consider the following least squares problem defined by

This translates to the following set of linear equations:

The corresponding least squares problem is

The preceding objective function can be expanded to

In addition, you impose the following constraint so that the equation is satisfied within a tolerance of 0.1:

You can create the QPS-format input data set by using the following SAS statements:

data mycas.lsdata;
   input _id_ field1 $ field2 $ field3 $ field4 field5 $ field6 @;
   datalines;
1  NAME     .      LEASTSQ     .            .         .
2  ROWS     .      .           .            .         .
3  N        OBJ    .           .            .         .
4  G        EQ3    .           .            .         .
5  COLUMNS  .      .           .            .         .
6  .        X1     OBJ        -14           EQ3       3
7  .        X2     OBJ        -4            EQ3       2
8  RHS      .      .           .            .         .
9  .        RHS    OBJ        -2            EQ3       0.9
10 RANGES   .      .           .            .         .
11 .        RNG    EQ3         0.2          .         .
12 BOUNDS   .      .           .            .         .
13 FR       BND1   X1          .            .         .
14 FR       BND1   X2          .            .         .
15 QUADOBJ  .      .           .            .         .
16 .        X1     X1          52           .         .
17 .        X1     X2          10           .         .
18 .        X2     X2          10           .         .
19 ENDATA   .      .           .            .         .
;

The decision variables and are free, so they have bound type FR in the BOUNDS section of the QPS-format data set.

You can use the following SAS statements to solve the least squares problem:

proc optqp data = mycas.lsdata
   printlevel = 0
   primalout = mycas.lspout;
run;

The optimal solution is displayed in Output 7.1.1.

The iteration log is shown in Output 7.1.2.