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 not clear at present. Jean-Baptiste Hiriart-Urruty posted it as an open question in the field of nonlinear analysis and optimization [‘Question 11’ in SIAM Review 49 (2007), 255-273]. 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. Furthermore, we show that the conjugate can be explicitly expressed as a single function in some situations. Our analysis shows that the relationship between the matrices of quadratic forms plays a vital role in determining whether or not the conjugate can be expressed explicitly, and our analysis also sheds some light on the computational complexity of the Legendre-Fenchel conjugate for the product of quadratic forms.