What are the advantages of a hyperbolic program over a semi definite program? SDPs can be used to represent a wide variety of algebraic constraints. Are there constraints that can be represented in a hyperbolic program but not a semi definite program? 


Is there a reference that describes an application or applications of hyperbolic programming? 


Has anyone developed a hyperbolic programming solver? If so, is there a reference describing its implementation? 



Advantages of hyperbolic programming over semidefinite programming?
*Disclaimer: I'm not an expert in the area, just a fellow curious.*


Update (2023): In the pre-print "[Sums of Squares Representations on Singular Loci](https://arxiv.org/abs/2303.05081)" by Ngoc Hoang Anh Mai and Victor Magron is claimed that every hyperbolic program is equivalent to a semidefinite program.




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* As far I known, it is still unknown if hyperbolic feasibility problems are polynomial-time equivalent to LMI problems (1).
* Moreover (as of October 2018), it is also still unknown if every hyperbolic program might be recast as a semidefinite program, possibly involving more variables (2).
* Finally, an algorithm for solving hyperbolic programs is proposed in (3), but I don't know if an implementation is available. You may try to contact the author of the paper.


(1) *Tunçel, Levent*, Polyhedral and semidefinite programming methods in combinatorial optimization, Fields Institute Monographs 26. Providence, RI: American Mathematical Society (AMS); Toronto: The Fields Institute for Research in Mathematical Sciences (ISBN 978-0-8218-3352-0/hbk). x, 219 p. (2010). [ZBL1207.90005](https://zbmath.org/?q=an:1207.90005).


(2) *Saunderson, James*, [**A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone**](http://dx.doi.org/10.1007/s11590-018-1246-x), Optimization Letters. 12, No. 7, 1475-1486 (2018).


(3) *Renegar, James*, [**Accelerated first-order methods for hyperbolic programming**](http://dx.doi.org/10.1007/s10107-017-1203-y), Math. Program., Ser. A (2017).


