ON CONDITIONALLY DEFINITE QUADRATIC FORMS

Abstract

ABSTRACT Second order optimality conditions may require the verification of a conditional definiteness condition of the form where Q(·) is a quadratic form on Rn and A is an m x n matrix. The prototype case, apart from that of positive definiteness, is that in which A is the n x n identity matrix—in this case the conditionally definite forms are called strictly copositive forms. It has recently been shown by the authors, using a theorem of Finsler on quadratic forms, that the general case can be reduced to that of strict copositivity, in that (≠) holds if and only if there exist a strictly copositive from C(·) and a positive definite form S(·) such that The present paper presents an entirely different treatment of this equivalence, in which (≠) is generalised to a condition of the form Γ being any given closed, but not necessarily convex, cone in Rm, while in (≠≠) the form C is required to be strictly positive on Γ. The equivalence is established by showing that for a fixed matrix A, the two convex cones ...