There is the concept of *hyperbolic programming*, introduced in the 90's by Osman Güler (*Hyperbolic Hyperbolic Polynomials and Interior Point Methods for Convex Programming*, Math. Oper. Res. **22** (1997) 350-377. See also H.H. Bauschke, O. Güler, A.S. Lewis, H.S. Sendov, *Hyperbolic Polynomials and Convex Analysis*, Canad. J. Math. **53** (2001) 470-488), based on the concept of *hyperbolicity cone* introduced by Lars Garding in the 50's in the context of the theory of hyperbolic partial differential equations.

We say that a polynomial $P:\mathbb{R}^n\rightarrow\mathbb{R}$ is *hyperbolic* with respect to $\tau\in\mathbb{R}^n$ if $P(\tau)\neq 0$ and the roots of the one-variable polynomial $P_{\xi,\tau}(\lambda)\doteq P(\xi-\lambda\tau)$ are all *real* for all $\xi\in\mathbb{R}^n$. We then define the *hyperbolicity cone* $C(P,\tau)$ of $P$ with respect to $\tau$ as $$C(P,\tau)=\{\xi\in\mathbb{R}^n\ |\ \text{the roots of }P_{\xi,\tau}\text{ are all positive}\}\ .$$ It was shown by Garding that $C(P,\tau)$ is an open, convex cone which is the connected component of $\{\xi\ |\ P(\xi)\neq 0\}$ to which $\tau$ belongs. Hyperbolic programming is then simply conic programming when the feasibility cone is an hyperbolicity cone of some hyperbolic polynomial.

Hyperbolic programming generalizes semidefinite programming in the following sense: if $\gamma_1,\ldots,\gamma_n$ are $N\times N$ symmetric matrices and we set $\gamma(\xi)=\sum^n_{j=1}\xi_j\gamma_j$, $\xi\in\mathbb{R}^n$, then $P(\xi)=\det\gamma(\xi)$ is an hyperbolic polynomial with respect to any $\tau\in\mathbb{R}^n$ such that $\gamma(\tau)$ is positive definite. It is not known whether this is a *strict* generalization of semidefinite programming, though - it is in fact conjectured that any hyperbolicity cone can be represented as a cone of positive definite matrices. Since this was originally conjectured (in a stronger form) by Peter Lax in the 50's for $3\times 3$ matrices, this (still open) conjecture became known as the *generalized Lax conjecture*. Lax's original claim was proven by A.S. Lewis, P.A. Parrilo and M.V. Ramana (*The Lax Conjecture is True*, Proc. Amer. Math. Soc. **133** (2005) 2495-2499) building on work by J.W. Helton and V. Vinnikov (Commun. Pure Appl. Math. **60** (2007) 654-674).