Constructing self-concordant barriers for convex cones

In this paper we develop a technique for constructing self-concordant barriers for convex cones. We start from a simple proof for a variant of standard result [1] on transformation of a -self-concordant barrier for a set into a self-concordant barrier for its conic hull with parameter (3.08 + 3.57)2 . Further, we develop a convenient composition theorem for constructing barriers directly for convex cones. In particular, we can construct now good barriers for several interesting cones obtained as a conic hull of epigraph of a univariate function. This technique works for power functions, entropy, logarithm and exponent function, etc. It provides a background for development of polynomial-time methods for separable optimization problems. Thus, our abilities in constructing good barriers for convex sets and cones become now identical.