Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or nice) which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured c
losed convex cones that contain all spectrahedra (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedra are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.
We construct a general framework for deriving error bounds for conic feasibility problems. In particular, our approach allows one to work with cones that fail to be amenable or even to have computable projections, two previously challenging barriers.
For the purpose, we first show how error bounds may be constructed using objects called facial residual functions. Then, we develop several tools to compute facial residual functions even in the absence of closed form expressions for the projections onto the cones. We demonstrate the use and power of our results by computing tight error bounds for the exponential cone feasibility problem. Interestingly, we discover a natural example for which the tightest error bound is related to the Boltzmann-Shannon entropy. We were also able to produce an example of sets for which a H{o}lderian error bound holds but the supremum of the set of admissible exponents is not itself an admissible exponent.
