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YunBin 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 LegendreFenchel 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 LegendreFenchel transform for it?} First, we show that the convexity of the product is determined intrinsically by the condition number of socalled `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 LegendreFenchel 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), 225273] is addressed in this paper. Keywords: Matrix analysis, convex analysis, LegendreFenchel transform, quadratic forms, positive definite matrices, condition numbers Category 1: Convex and Nonsmooth Optimization Category 2: Nonlinear Optimization Citation: Download: [PDF] Entry Submitted: 04/24/2010 Modify/Update this entry  
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