- Convexity Conditions and the Legendre-Fenchel Transform for the Product of Finitely Many Positive Definite Quadratic Forms Yun-Bin Zhao(zhaoyymaths.bham.ac.uk) Abstract: While the product of finitely many convex functions has been investigated in the field of global optimization, some fundamental issues such as the convexity condition and the Legendre-Fenchel transform for the product function remain unresolved. Focusing on quadratic forms, this paper is aimed at addressing the question: \emph{When is the product of finitely many positive definite quadratic forms convex, and what is the Legendre-Fenchel transform for it?} First, we show that the convexity of the product is determined intrinsically by the condition number of so-called scaled matrices' associated with quadratic forms involved. The main result claims that if the condition number of these scaled matrices are bounded above by an explicit constant (which depends only on the number of quadratic forms involved), then the product function is convex. Second, we prove that the Legendre-Fenchel transform for the product of positive definite quadratic forms can be expressed, and the computation of the transform amounts to finding the solution to a system of equations (or equally, finding a Brouwer's fixed point of a mapping) with a special structure. Thus, a broader question than the open `Question 11" in [\emph{SIAM Review}, 49 (2007), 225-273] is addressed in this paper. Keywords: Matrix analysis, convex analysis, Legendre-Fenchel transform, quadratic forms, positive definite matrices, condition numbers Category 1: Convex and Nonsmooth Optimization Category 2: Nonlinear Optimization Citation: Download: [PDF]Entry Submitted: 04/24/2010Entry Accepted: 04/24/2010Entry Last Modified: 04/24/2010Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.