The source files for all examples can be found in /examples.

Closest Correlation Matrix

We consider the problem of finding the closest correlation matrix $X$ to a given random matrix $C$. With closest correlation matrix we mean a positive semidefinite matrix with ones on the diagonal. The problem is given by:

\[\begin{array}{ll} \text{minimize} & \frac{1}{2}||X - C||_F^2\\ \text{subject to} & X_{ii} = 1, \quad i=1,\dots,n \\ & X \succeq 0. \end{array}\]

Notice that we use JuMP to model the problem. COSMO is chosen as the backend solver. And COSMO-specific settings are passed using the optimizer_with_attributes() function.

using COSMO, JuMP, LinearAlgebra, SparseArrays, Test, Random

rng = Random.MersenneTwister(12345);
# create a random test matrix C
n = 8;
C = -1 .+ rand(rng, n, n) .* 2;
c = vec(C);

Define problem in JuMP:

q = -vec(C);
r = 0.5 * vec(C)' * vec(C);
m = JuMP.Model(optimizer_with_attributes(COSMO.Optimizer, "verbose" => true));
@variable(m, X[1:n, 1:n], PSD);
x = vec(X);
@objective(m, Min, 0.5 * x' * x  + q' * x + r);
for i = 1:n
  @constraint(m, X[i, i] == 1.);
end

Solve the JuMP model with COSMO and query the solution X_sol:

status = JuMP.optimize!(m);
obj_val = JuMP.objective_value(m);
X_sol = JuMP.value.(X);

------------------------------------------------------------------
          COSMO v0.8.9 - A Quadratic Objective Conic Solver
                         Michael Garstka
                University of Oxford, 2017 - 2022
------------------------------------------------------------------

Problem:  x ∈ R^{36},
          constraints: A ∈ R^{44x36} (44 nnz),
          matrix size to factor: 80x80,
          Floating-point precision: Float64
Sets:     DensePsdConeTriangle of dim: 36 (8x8)
          ZeroSet of dim: 8
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 = 15, RestartedMemory,
          Safeguarded: true, tol: 2.0
Setup Time: 963.88ms

Iter:	Objective:	Primal Res:	Dual Res:	Rho:
1	 9.8585e+00	5.9778e-01	7.6572e-01	1.0000e-01
25	 2.2101e+00	4.4633e-07	7.3877e-08	1.0000e-01

------------------------------------------------------------------
>>> Results
Status: Solved
Iterations: 25
Optimal objective: 2.21
Runtime: 5.201s (5201.02ms)

Double check result against known solution:

known_opt_val = 12.5406
known_solution =  [
  1.0         0.732562   -0.319491   -0.359985   -0.287543   -0.15578     0.0264044  -0.271438;
  0.732562    1.0         0.0913246  -0.0386357   0.299199   -0.122733    0.126612   -0.187489;
 -0.319491    0.0913246   1.0        -0.0863377   0.432948    0.461783   -0.248641   -0.395299;
 -0.359985   -0.0386357  -0.0863377   1.0         0.503379    0.250601    0.141151    0.286088;
 -0.287543    0.299199    0.432948    0.503379    1.0        -0.0875199   0.137518    0.0262425;
 -0.15578    -0.122733    0.461783    0.250601   -0.0875199   1.0        -0.731556    0.0841783;
  0.0264044   0.126612   -0.248641    0.141151    0.137518   -0.731556    1.0        -0.436274;
 -0.271438   -0.187489   -0.395299    0.286088    0.0262425   0.0841783  -0.436274    1.0  ];
@test isapprox(obj_val, known_opt_val , atol=1e-3)

Test Passed

@test norm(X_sol - known_solution, Inf) < 1e-3

Test Passed

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