Error bounds, facial residual functions and applications to the exponential cone

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Abstract
We construct a general framework that can be used to derive 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 impassable barriers. For the purpose, we first show how error bounds may be constructed using objects called facial residual functions. We then show how facial residual functions may be explicitly computed by checking verifiable limits of sequences whose forms do not depend upon projections onto the cones. We show how to apply these methods with chains of faces, and arbitrary products of such cones. We demonstrate the use and power of our results by completely determining the elusive error bounds for the exponential cone feasibility problem, obtaining all of its facial residual functions. Interestingly, we discover a natural example for which the resulting H\"{o}lderian error bound exponents form a set whose supremum is not itself an admissible exponent.

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