ILOG CPLEX 11.0 User's Manual

convex QPs, Q must be positive semi-definite; that is, x T Qx ≥ 0 for every vector x, whetheror not x is feasible. For concave maximization problems, the requirement is that Q must benegative semi-definite; that is, x T Qx ≤ 0 for every vector x. It is conventional to use the sameterm, positive semi-definite, abbreviated PSD, for both cases, on the assumption that amaximization problem with a negative semi-definite Q can be transformed into an equivalentPSD.For a separable function, it is sufficient to check whether the individual diagonal elements ofthe matrix Q are of the correct sign. For a nonseparable case, it may be less easy to decide inadvance the convexity of Q. However, ILOG CPLEX detects this property during the earlystages of optimization and terminates if the quadratic objective term in a QP is found to benot PSD.For a more complete explanation of quadratic programming generally, a text, such as one ofthose listed in Further Reading on page 39 of the preface of this manual, will be helpful.Entering QPsILOG CPLEX supports two views of quadratic objective functions: a matrix view and analgebraic view.◆ Matrix View on page 229◆ Algebraic View on page 230◆ Examples for Entering QPs on page 230◆ Reformulating QPs to Save Memory on page 231Matrix ViewIn the matrix view, commonly found in textbook presentations of QP, the objective functionis defined as 1/2 x T Qx+c T x, where Q must be symmetric and positive semi-definite for aminimization problem, or negative semi-definite for a maximization problem. This view issupported by the MPS file format and the Callable Library routines, where the quadraticobjective function information is specified by providing the matrix Q. Thus, by definition,the factor of 1/2 must be considered when entering a model using the matrix view, as it willbe implicitly assumed by the optimization routines.Similarly, symmetry of the Q matrix data is required; the MPS reader will return an errorstatus code if the file contains unequal off-diagonal components, such as a nonzero value forone and zero (or omitted) for the other.This symmetry restriction applies to quadratic programming input formats rather than thequadratic programming problem itself. For models with an asymmetric Q matrix, eitherexpress the quadratic terms algebraically, as described in Algebraic View on page 230, orILOG CPLEX 11.0 — USER’ S MANUAL 229