Primal-dual path-following algorithms for circular programming

Circular programming problems are a new class of convex optimization problems in which we minimize linear function over the intersection of an affine linear manifold with the Cartesian product of circular cones. It has very recently been discovered that, unlike what has previously been believed, circular programming is a special case of symmetric programming, where it lies between second-order cone programming and semidefinite programming. Monteiro [SIAM J. Optim. 7 (1997) 663–678] introduced primal-dual path-following algorithms for solving semidefinite programming problems. Alizadeh and Goldfarb [Math. Program. Ser. A 95 (2003) 3–51] introduced primal-dual path-following algorithms for solving second-order cone programming problems. In this paper, we utilize their work and use the machinery of Euclidean Jordan algebras to derive primal-dual path-following interior point algorithms for circular programming problems. We prove polynomial convergence of the proposed algorithms by showing that the circular logarithmic barrier is a strongly self-concordant barrier.