ILOG CPLEX distinguishes between a quadratic algebraic term and a quadratic matrix coefficient. The quadratic algebraic terms are the coefficients that appear in the algebraic expression defined as part of the ILOG CPLEX LP format. The quadratic matrix coefficients appear in Q. The quadratic coefficient of an off-diagonal term must be distributed within the Q matrix, and it is always one-half the value of the quadratic algebraic term.

To clarify that terminology, consider this example:

Minimize a + b  + 1/2(a2 + 4ab + 7b2)

subject to a + b  10

with these bounds a  0 and b  0

The off-diagonal quadratic algebraic term in that example is 4, so the quadratic matrix Q is

To change the off-diagonal quadratic term from 4 to 6, use this sequence of commands in the Interactive Optimizer:

CPLEX> change qpterm
Change which quadratic term ['variable' 'variable']: a b
Present quadratic term of variable 'a', variable 'b' is 4.000000.
Change quadratic term of variable 'a', variable 'b' to what: 6.0
Quadratic term of variable 'a', variable 'b' changed to 6.000000.

From the Callable Library, the CPXchgqpcoef call to change the off-diagonal term from 4 to 6 would change both of the off-diagonal matrix coefficients from 2 to 3. Thus, the indices would be 0 and 1, and the new matrix coefficient value would be 3.

If you have entered a linear problem without any quadratic terms, and you want to create quadratic terms, you must first change the problem type to QP. To do so, use the command change problem qp. This command will create an empty quadratic matrix with Q = 0.

When you change quadratic terms, there are still restrictions on the properties of the Q matrix. In a minimization problem, it must be convex, positive semi-definite. In a maximization problem, it must be concave, negative semi-definite. For example, if you change the sense of an objective function in a convex Q matrix from minimization to maximization, you will thus make the problem unsolvable. Likewise, in a convex Q matrix, if you make a diagonal term negative, you will thus make the problem unsolvable.