Tight bounds on Lyapunov rank

Abstract

The Lyapunov rank of a cone is the number of independent equations obtainable from an analogue of the complementary slackness condition in cone programming problems, and more equations are generally thought to be better. Bounding the Lyapunov rank of a proper cone in $$ \mathbb {R}^{n}$$ from above has been an open problem. Gowda and Tao gave an upper bound of $$n^{2} - n$$ that was later improved by Orlitzky and Gowda to $$ \left( {n-1}\right) ^{2}$$ . We settle the matter and show that the Lyapunov rank of $$ \left( {n^{2} - n}\right) /2 + 1$$ corresponding to the Lorentz cone is maximal.