Arbitrary Precision

COSMO allows you to solve problems with arbitrary floating-point precision, e.g. by using BigFloat problem data. To do this, the desired floating point type has to be consistent across the model COSMO.Model{<: AbstractFloat}, the input data and the (optional) settings object COSMO.Settings{<: AbstractFloat}. As an example, assume we want to solve the following quadratic program:

\[\begin{array}{ll} \text{minimize} & 1/2 x^\top P x + q^\top x \\ \text{subject to} & l \leq A x \leq u \end{array}\]

where $P = \begin{bmatrix} 4 & 1 \\ 1 & 2\end{bmatrix}$, $q = [1, 1]^\top$, $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ 0 & 1\end{bmatrix}$ and $l= [1,0, 0]^\top$, $u=[1, 0.7, 0.7]^\top$. We start by creating the model with the desired precision:

using COSMO, LinearAlgebra, SparseArrays
model = COSMO.Model{BigFloat}()

A COSMO Model with Float precision: BigFloat

Next, we define the problem data as BigFloat arrays and create the constraint:

q = BigFloat[1; 1.]
P = sparse(BigFloat[4. 1; 1 2])
A = BigFloat[1. 1; 1 0; 0 1]
l = BigFloat[1.; 0; 0]
u = BigFloat[1; 0.7; 0.7]

constraint = COSMO.Constraint(A, zeros(BigFloat, 3), COSMO.Box(l, u))

Constraint{BigFloat}
Size of A: (3, 2)
ConvexSet: COSMO.Box{BigFloat}

Notice that the constraint type parameter is dependent on the input data. The same is true for the constraint set Box. Next, we define the settings

settings = COSMO.Settings{BigFloat}(verbose = true, kkt_solver = QdldlKKTSolver)

A COSMO.Settings{BigFloat} object. To list the available options type `?` and `help?>COSMO.Settings`.

and assemble and solve the problem:

assemble!(model, P, q, constraint, settings = settings)
result = COSMO.optimize!(model);

------------------------------------------------------------------
          COSMO v0.8.8 - A Quadratic Objective Conic Solver
                         Michael Garstka
                University of Oxford, 2017 - 2022
------------------------------------------------------------------

Problem:  x ∈ R^{2},
          constraints: A ∈ R^{3x2} (4 nnz),
          matrix size to factor: 5x5,
          Floating-point precision: BigFloat
Sets:     Box of dim: 3
Settings: ϵ_abs = 1.0e-05, ϵ_rel = 1.0e-05,
          ϵ_prim_inf = 1.0e-04, ϵ_dual_inf = 1.0e-04,
          ρ = 0.1, σ = 1e-06, α = 1.6,
          max_iter = 5000,
          scaling iter = 10 (on),
          check termination every 25 iter,
          check infeasibility every 40 iter,
          KKT system solver: QDLDL
Acc:      Anderson Type2{QRDecomp},
          Memory size = 5, RestartedMemory,
          Safeguarded: true, tol: 2.0
Setup Time: 1313.67ms

Iter:	Objective:	Primal Res:	Dual Res:	Rho:
1	-7.8808e-03	1.0079e+00	2.0033e+02	1.0000e-01
25	 1.8800e+00	1.2972e-77	7.3280e-77	1.0000e-01

------------------------------------------------------------------
>>> Results
Status: Solved
Iterations: 25
Optimal objective: 1.88
Runtime: 3.201s (3200.56ms)

Moreover, notice that when no type parameter is specified, all objects default to Float64:

model = COSMO.Model()

A COSMO Model with Float precision: Float64

Two limitations to arbitrary precision:

• Since we call the LAPACK function syevr for eigenvalue decompositions, we currently only support solving problems with PSD constraints in Float32 and Float64.
• We suggest to use the pure Julia QDLDL linear system solver (kkt_solver = QdldlKKTSolver) when working with arbitrary precision types as some of the other available solvers don't support all available precisions.

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