A note on Legendre-Fenchel conjugate of the product of two positive-definite quadratic forms

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 {\it SIAM Review} 49, 255-273, (2007)]. Under a convex assumption on the function, it was answered by Zhao [SIAM J. Matrix Analysis $\&$ Applications, 31(4), … Read more

Separation and Relaxation for cones of quadratic forms

Let P be a pointed, polyhedral cone in R_n. In this paper, we study the cone C = cone{xx^T: x \in P} of quadratic forms. Understanding the structure of C is important for globally solving NP-hard quadratic programs over P. We establish key characteristics of C and construct a separation algorithm for C provided one … Read more

The Legendre-Fenchel Conjugate of the Product of Two positive definite Quadratic Forms

It is well-known that the Legendre-Fenchel 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 inf-convolution. However, the Legendre-Fenchel conjugate of the product of two positive definite quadratic forms is … Read more