A Note on Legendre-Fenchel Conjugate of the Product of Two Positive-Definite Quadratic Forms

Abstract

The Legendre-Fenchel conjugate of the product of two positive-definite quadratic forms was posted as an open question in the field of nonlinear analysis and optimization by Hiriart-Urruty ['Question 11' in SIAM Review 49, 255-273, 2007]. Under a convexity assumption on the function, it was answered by Zhao [SIAM J. Matrix Analysis & Applications31(4), 1792-1811, 2010]. In this note, we answer the open question without making the convexity assumption. © 2013 Operations Research Society of China, Periodicals Agency of Shanghai University, and Springer-Verlag Berlin Heidelberg.