Multivariate sharp quadratic bounds via $\mathbf{\Sigma}$-strong convexity and the Fenchel connection

Sharp majorization is extended to the multivariate case. To achieve this, the notions of $\sigma$-strong convexity, monotonicity, and one-sided Lipschitz continuity are extended to $\mathbf{\Sigma}$-strong convexity, monotonicity, and Lipschitz continuity, respectively. The connection between a convex function and its Fenchel-Legendre transform is then developed. Sharp majorization is illustrated in single and multiple dimensions, and we show …