The volumetric barrier for convex quadratic constraints

Let Q = fx j s i (x) 0; i = 1; : : : ; kg, where s i (x) = a T i x ?x T Q i x ?c i , and Q i is an n n positive semideenite matrix. We prove that the volumetric and combined volumetric-logarithmic barriers for Q are O(p kn) and O(p kn) self-concordant, respectively. Our analysis uses the semideenite programming (SDP) representation for the convex quadratic constraints deening Q, and our earlier results on the volumetric barrier for SDP. The self-concordancy results actually hold for a class of SDP problems more general than those corresponding to the SDP representation of Q.