The distinguishing characteristic of QCP is that quadratic terms may appear in one or more constraints of the problem. The objective function of such a problem may or may not contain quadratic terms as well. Thus, the most general formulation of a QCP is:

Minimize    1/2xTQx + cTx

subject to   Ax ~ b

and            aiTx + xTQix ri for i=1,...,q

with these bounds l  x  u

As with a quadratic objective function, convexity plays an important role in quadratic constraints. The constraints must each define a convex region. To make sure of convexity, ILOG CPLEX requires that each Qi matrix be positive semi-definite (PSD) or that the constraint must be in the form of a second order cone. The following sections offer more information about these concepts.