A Condition for Convexity of a Product of Positive Definite Quadratic Forms

A sufficient condition for the convexity of a finite product of positive definite quadratic forms is given in terms of the condition numbers of the underlying matrices. When only two factors are involved the condition is also necessary. This complements and improves a result recently obtained by Zhao [Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms, Applied Mathematics and Optimization, Volume 62, (2010) Number 3, 411-434]. As a special case, a necessary and sufficient condition is given for the Kantorovich function (xT Ax)(xT A−1x), where A is positive definite, to be convex. Key words. Legendre-Fenchel transform, quadratic form, positive definite matrix, condition number. AMS subject classifications. 90C25, 65K05, 65F15, 15A48

• Type: Journal
• Year: 2011
• Where: SIAMMAX
• Authors: Minghua Lin, Gord Sinnamon
• Keywords: Condition Numbers, Convexity, Quadratic Form, SIAMMAX 2011