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Yunbin Zhao (zhaoyymaths.bham.ac.uk) Abstract: It is wellknown that the LegendreFenchel conjugate of a positive definite quadratic form can be explicitly expressed as another positive definite quadratic form, and that the conjugate of the sum of several positive definite quadratic forms can be expressed via infconvolution. However, the LegendreFenchel conjugate of the product of two positive definite quadratic forms is not clear at present. JeanBaptiste HiriartUrruty posted it as an open question in the field of nonlinear analysis and optimization [`Question 11' in \emph{SIAM Review} 49 (2007), 255273]. From convex analysis point of view, it is interesting and important to address such a question. The purpose of this paper is to answer this question and to provide a formula for the conjugate of the product of two positive definite quadratic forms. We prove that the computation of the conjugate can be implemented via finding a root to certain univariate polynomial equation, and we also identify the situations in which the conjugate can be explicitly expressed as a single function without involving any parameter. Some other issues, including the convexity condition for the product function, are also investigated as well. Our analysis shows that the relationship between the matrices of quadratic forms plays a vital role in determining whether the conjugate can be explicitly expressed or not. Keywords: Convex analysis, matrix theory, quadratic form, LegendreFenchel conjugate Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Download: [PDF] Entry Submitted: 09/19/2009 Modify/Update this entry  
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