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.