Linear System Solver
One major step of COSMO's ADMM algorithm is solving a linear system of equations at each iteration. Fortunately, the left-hand matrix is only dependent on the problem data and therefore only needs to be factored once. Depending on the problem class this factorisation can be the computationally most expensive step of the algorithm (LPs, QPs). See the Method section for a more detailed description of the linear system.
COSMO allows you to specify the linear system solver that performs the factorisation and back-substitution. We also support indirect system solver which are useful for very large problems where a factorisation becomes inefficient. The table below shows the currently supported linear system solver:
| Type | Solver | Description |
|---|---|---|
| CholmodKKTSolver | Cholmod | Julia's default linear system solver (from SuiteSparse) |
| QdldlKKTSolver | QDLDL | For more information QDLDL.jl |
| PardisoDirectKKTSolver | Pardiso (direct) | Pardiso 6.0 direct solver |
| PardisoIndirectKKTSolver | Pardiso (indirect) | Pardiso 6.0 indirect solver |
| MKLPardisoKKTSolver | Intel MKL Pardiso | Pardiso optimised for Intel platforms |
| CGIndirectKKTSolver | IterativeSolvers.jl | Conjugate Gradients on the reduced KKT linear system. |
| MINRESIndirectKKTSolver | IterativeSolvers.jl | MINRES on the (full) KKT linear system. |
To use the Pardiso and Intel MKL Pardiso solver, you have to install the respective libraries and the corresponding Julia wrapper. For more information about installing these, visit the Pardiso.jl repository page. Likewise in order to use Indirect(Reduced)KKTSolver you have to install IterativeSolvers.jl (v0.9+) and LinearMaps.jl. We are using the Requires package for lazy loading of code related to Pardiso and IterativeSolvers. This means in order to use Pardiso / IterativeSolvers, you'll have to load these packages alongside COSMO, i.e. using Pardiso and using IterativeSolvers, LinearMaps.
COSMO uses the Cholmod linear system solver by default. You can specify a different solver in the settings by using the kkt_solver keyword and the respective type:
settings = COSMO.Settings(kkt_solver = CholmodKKTSolver)
COSMO also allows you to pass in solver-specific options with the with_options(solver_type, args...; kwargs...) syntax. For example, if you want to use Pardiso with verbose printing use the following syntax:
settings = COSMO.Settings(kkt_solver = with_options(PardisoDirectKKTSolver, msg_level_on = true))Likewise, CGIndirectKKTSolver and MINRESIndirectKKTSolver are also parameterizable with with_options(solver_type, args...; kwargs...) and accept the following arguments:
| Argument | Description | Values (default) |
|---|---|---|
tol_constant::T and tol_exponent::T | Parameter that defines the solution tolerance for the iterative solvers across iterations. In particular, the solution tolerance at every iteration is defined as $\text{tol\_constant} / \text{iteration}^{\text{tol\_exponent}}$ | 1.0, 1.5 |
This also works if you want to use this configuration with JuMP:
model = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "kkt_solver" => with_options(PardisoDirectKKTSolver, msg_level_on = true));
Or alternatively:
model = JuMP.Model(COSMO.Optimizer);
set_optimizer_attribute(model, "kkt_solver", with_options(PardisoDirectKKTSolver, msg_level_on = true));