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


*

*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.
(2) Saunderson, James, A spectrahedral representation of the first derivative relaxation of the positive semidefinite cone, Optimization Letters. 12, No. 7, 1475-1486 (2018).
(3) Renegar, James, Accelerated first-order methods for hyperbolic programming, Math. Program., Ser. A (2017).
