Copositive Matrices and Definiteness of Quadratic Forms Subject to Homogeneous Linear Inequality Constraints

摘要

A symmetric matrix C is called copositive if the quadratic form x prime C x is nonnegative for all nonnegative values of the variables (x sub 1, x sub 2,...., x sub n)= x prime o. A known sufficient condition for a quadratic from x prime Qx to be positive unless x=0, subject to the linear inequality constraints Ax equal to or greater than 0, is that there should exist a copositive matrix C such that Q - A prime CA is positive definite. The main result establishes the necessity of this condition. For x prime Qx to be merely nonnegative subject to Ax equal to or greater than 0, the situation is less straight-forward. The necessity of the existence of a copositive matrix C such that Q - A prime CA is positive semi-definite is proved only under various additional hypotheses regarding the size or rank of A, and counter examples are given to show that, in general, no such matrix may exist, even when Slater's constraint qualification holds. Our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity in which specific symmetric matrices, constructed by rational operations from A and Q and depending upon a single real parameter nu, must be tested for positive definiteness or strict copositivity for large values of nu. This technique is illustrated by several examples.