COSMO: A Conic Operator Splitting Method for Convex Conic Problems

• Published in: Journal of optimization theory and applications Vol. 190; no. 3; pp. 779 - 810
• Main Authors: Garstka, Michael, Cannon, Mark, Goulart, Paul
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2021; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Computational geometry; Convexity; Engineering; Graph theory; Iterative methods; Mathematical analysis; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Operators (mathematics); Optimization; Robust control; Solvers; Splitting; Theory of Computation; 49K99; More; 49J53; Clique merging; Chordal decomposition; ADMM; Conic programming
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-021-01896-x

This paper describes the conic operator splitting method (COSMO) solver, an operator splitting algorithm and associated software package 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 low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to effectively exploit sparsity in large, structured semidefinite programs. Numerical comparisons with other state-of-the-art solvers for a variety of benchmark problems show the effectiveness of our approach. Our Julia implementation is open source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem.