A Condition Number for Second Order Conic Systems

In this note we define a condition number C (A) for the feasibility problem of homogeneous second order linear conic systems. This condition number extends that defined by the first two authors in Math. Program. 91:163–174 (2001) for homogeneous polyhedral linear conic systems. We show that it shares some of its properties —it can be characterized as a relativised inverse to the distance to ill-posedness, it is bounded by Renegar’s condition number C(A)— but not all of them —it is not necessarily the inverse of the radius of the largest cone inscribed in the solution set. In doing so, we also prove a new characterization of Renegar’s C(A).