COSMO: A conic operator splitting method for large convex problems

This paper describes the Conic Operator Splitting Method (COSMO), an operator splitting algorithm for convex optimisation problems with quadratic objective function and conic constraints. At each step the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The solver is able to exploit chordal sparsity in the problem data and to detect infeasible problems. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs in portfolio optimisation, graph theory, and robust control. Our Julia implementation is open-source, extensible, integrated into the Julia optimisation ecosystem and performs well on a variety of large convex problem classes.

I. Introduction

We consider convex optimisation problems in the form

$$\begin{align*} \text{minimize}\quad & f(x)\\ \text{subject to}\quad & g_{i}(x)\leq 0,\quad i=1,\ldots, l\\ & h_{i}(x)=0,\quad i=1, \ldots, k, \tag{1} \end{align*}$$ where we assume that both the objective function : and the inequality constraint functions : are convex, and that the equality constraints are affine. Convex optimisation problems feature in a wide range of applications, including problems in machine learning, finance, optimal control, and operations research [1]. Concrete examples of problems fitting the general form (1) include linear programming (LP), quadratic programming (QP), second-order cone programming (SOCP), and semidefinite programming (SDP) problems. Methods to solve each of these standard problem classes are well understood and a number of open-and closed-source solvers are widely available. However, the trend for data and training sets of increasing size in decision problems and machine learning poses a challenge for state-of-the-art software.