ILOG CPLEX 11.0 User's Manual

Identifying Convex QPsConventionally, a quadratic program (QP) is formulated this way:Minimize1 / 2 x T Qx+c T xsubject to Ax ~ bwith these bounds l ≤ x ≤ uwhere the relation ~ may be any combination of equal to, less than or equal to, greater thanor equal to, or range constraints. As in other problem formulations, l indicates lower and uupper bounds. Q is a matrix of objective function coefficients. That is, the elements Q jj arethe coefficients of the quadratic terms x 2 j , and the elements Q ij and Q ji are summed togetherto be the coefficient of the term x i x j .ILOG CPLEX distinguishes two kinds of Q matrices:◆In a separable problem, only the diagonal terms of the matrix are defined.◆ In a nonseparable problem, at least one off-diagonal term of the matrix is nonzero.ILOG CPLEX can solve minimization problems having a convex quadratic objectivefunction. Equivalently, it can solve maximization problems having a concave quadraticobjective function. All linear objective functions satisfy this property for both minimizationand maximization. However, you cannot always assume this property in the case of aquadratic objective function. Intuitively, recall that any point on the line between twoarbitrary points of a convex function will be above that function. In more formal terms, acontinuous segment (that is, a straight line) connecting two arbitrary points on the graph ofthe objective function will not go below the objective in a minimization, and equivalently,the straight line will not go above the objective in a maximization. Figure 12.1 illustrates thisintuitive idea for an objective function in one variable. It is possible for a quadratic functionin more than one variable to be neither convex nor concave.Figure 12.1Figure 12.1 Minimize a Convex Objective Function, Maximize a Concave Objective FunctionIn formal terms, the question of whether a quadratic objective function is convex or concaveis equivalent to whether the matrix Q is positive semi-definite or negative semi-definite. For228 ILOG CPLEX 11.0 — USER’ S MANUAL