alfonso: Matlab package for nonsymmetric conic optimization

Abstract

We present alfonso, an open-source Matlab package for solving conic optimization problems over nonsymmetric convex cones. The implementation is based on the authors' corrected analysis of a primal-dual interior-point method of Skajaa and Ye. This method enables optimization over any convex cone as long as a logarithmically homogeneous self-concordant barrier is available for the cone or its dual. This includes many nonsymmetric cones, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Besides enabling the solution of problems which cannot be cast as optimization problems over a symmetric cone, it also offers performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. The worst-case iteration complexity of alfonso is the best known for non-symmetric cone optimization: $O(\sqrt{\nu}\log(1/\epsilon))$ iterations to reach an $\epsilon$-optimal solution, where $\nu$ is the barrier parameter of the barrier function used in the optimization. alfonso can be interfaced with a Matlab function (supplied by the user) that computes the Hessian of a barrier function for the cone. For convenience, a simplified interface is also available to optimize over the direct product of cones for which a barrier function has already been built into the software. This interface can be easily extended to include new cones. Both interfaces are illustrated by solving linear programs. The oracle interface and the efficiency of alfonso are also demonstrated using a design of experiments problem in which the tailored barrier computation greatly decreases the solution time compared to using state-of-the-art conic optimization software.