If you use the ILOG CPLEX barrier optimizer for quadratic programming problems (QPs), then you can specify quadratic objective coefficients in MPS format in a QMATRIX section.

Following the BOUNDS section, a QMATRIX section may be specified. Each line of this section defines one nonzero coefficient of the matrix Q. Each line should contain two variable names (which must have been specified in the COLUMNS section) in Fields 2 and 3, followed by a nonzero coefficient value in Field 4. For each off-diagonal coefficient, two lines must appear: one for the lower triangular element, and one for the upper triangular element. ILOG CPLEX evaluates the quadratic part of the objective function as 0.5 x'Qx, when the coefficients of Q are specified in an MPS file.

For example, consider the following problem:

Minimize
	a	+	b	+	1/2	(a2	+	4ab	+	7b2)
subject to
	a	+	b		10
	a,	b		0

In MPS format, you may enter the problem in the following way:

NAME          problem
ROWS
 N  obj
 G  c1
COLUMNS
    a         obj                  1   c1                   1
    b         obj                  1   c1                   1
RHS
    rhs       c1                  10
QMATRIX
    a         a                    1
    a         b                    2
    b         a                    2
    b         b                    7
ENDATA

You can also enter the quadratic objective coefficients by using a QUADOBJ section. In this format, only the upper diagonal elements of the Q matrix are entered. For the same example, the input with a QUADOBJ section looks like this:

NAME          problem
ROWS
 N  obj
 G  c1
COLUMNS
    a         obj                  1   c1                   1
    b         obj                  1   c1                   1
RHS
    rhs       c1                  10
QUADOBJ
    a         a                    1
    a         b                    2
    b         b                    7
ENDATA

If you have a model with quadratic objective information in MPS format in a QUADOBJ section of the following form, you do not have to convert your file in order for ILOG CPLEX to make use of it.

varname1 varname2 value

ILOG CPLEX can read that file and interpret the QUADOBJ section correctly. However, the MPS file writers of ILOG CPLEX do not produce a QUADOBJ section themselves. Instead, they produce a QMATRIX section, as explained here.