Strong Convexity of Sandwiched Entropies and Related Optimization Problems

Abstract

We present several theorems on strict and strong convexity, and higher order differential formulae for sandwiched quasi-relative entropy (a parametrised version of the classical fidelity). These are crucial for establishing global linear convergence of the gradient projection algorithm for optimisation problems for these functions. The case of the classical fidelity is of special interest for the multimarginal optimal transport problem (the $n$-coupling problem) for Gaussian measures.

In one of the statements of Theorem 4 of the first version of this paper, an extra condition that is needed was missing. This has been corrected in the present version, where that Theorem is now Theorem 5. The rest of the paper is unaffected