There is the concept of *hyperbolic programming*, introduced in the 90's by Osman Güler: 

 - O. Güler, *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;
- J. Renegar, *Hyperbolic Programs, and their Derivative Relaxations*, Found. Comput. Math. **6** (2005) 59–79. 

It is based on the concept of *hyperbolicity cone* introduced by Lars Gårding in the 50's in the context of the theory of hyperbolic partial differential equations.

We say that an homogeneous 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 Gårding that $C(P,\tau)$ is an open, convex cone which is the connected component of $\{\xi\ |\ P(\xi)\neq 0\}$ to which $\tau$ belongs. Moreover, the closure of $C(P,\tau)$ has the form 
$$
\overline{C(P,\tau)}=\{\xi\in\mathbb{R}^n\ |\ \text{the roots of }P_{\xi,\tau}\text{ are all nonnegative}\}\ .
$$ 
Hyperbolic programming is then simply conic programming when the feasibility cone is (the closure of) an hyperbolicity cone of some hyperbolic polynomial $P$. Checking if $\xi\in\mathbb{R}^n$ belongs to $C(P,\tau)$ (resp. $\overline{C(P,\tau)}$) amounts to checking if the coefficients of the monic one-variable polynomial $P(\tau)^{-1}P_{\xi,\tau}=P_{\xi,\tau}(0)^{-1}P_{\xi,\tau}$ are all positive (resp. nonnegative).

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. In that case, 
$$
C(P,\tau)=\{\xi\in\mathbb{R}^n\ |\ \gamma(\xi)\text{ is positive definite}\}
$$ 
and 
$$
\overline{C(P,\tau)}=\{\xi\in\mathbb{R}^n\ |\ \gamma(\xi)\text{ is positive semidefinite}\}\ .
$$ 
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 for a suitable choice of $N$. Since this was originally conjectured (in a stronger form) by Peter Lax in the 50's for $n=3$, 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 (*Linear Matrix Inequality Representation of Sets*, Commun. Pure Appl. Math. **60** (2007) 654-674).