Abstract
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.
Highlights
We consider convex optimisation problems in the form minimise f (x) subject to gi (x) Ki 0, i = 1, . . . , l (1)hi (x) = 0, i = 1, . . . , k, with proper convex cones Ki ⊆ Rmi and where we assume that both the objective function f : Rn → R and the inequality constraint functions gi : Rmi → R are convex, and that the equality constraints hi (x) := ai x − bi are affine
We have implemented our algorithm in the conic operator splitting method (COSMO), an open-source package written in Julia [9]
The clique graph-based merging strategy with COSMO is compared to MOSEK and SCS on large structured semidefinite programming (SDP) from the SDPLib benchmark set [10], non-chordal SDPs generated with sparsity patterns from the SuiteSparse Matrix Collections [18], and SDPs with block-arrow shaped sparsity pattern
Summary
We describe a first-order method for large conic problems that directly supports quadratic objective functions, reducing overheads for applications with both quadratic objective function and PSD constraints This avoids a major disadvantage of conic solvers compared to native QP solvers, i.e. no additional matrix factorisation for the conversion is needed and favourable sparsity in the objective can be maintained. We show that our new clique graph-based merging strategy reduces the projection time of the algorithm by up to 60% in benchmark tests The combination of this decomposition strategy with a multithreaded projection step allows us to solve very large sparse and structured problems orders of magnitudes faster than competing solvers.
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