Defining quantum divergences via convex optimization

We introduce a new quantum R\'enyi divergence $D^{\#}_{\alpha}$ for $\alpha \in (1,\infty)$ defined in terms of convex optimization program. This has several desirable computational and operational properties such as an efficient semidefinite programming representation states channels, chain rule property. An important property this is that its regularization equal to the sandwiched (also known minimal) divergence. allows us prove results. First, we use it get converging hierarchy upper bounds on regularized $\alpha$-R\'enyi between channels > 1$. Second 1$ which characterize strong converse exponent channel discrimination. Finally improved capacities.