SOCP Example in Julia
We will show how to solve this problem both natively in Clarabel.jl and also by solving with Clarabel.jl within JuMP.
Clarabel.jl native interface
using Clarabel, SparseArrays
P = sparse([0. 0.;0. 1.].*2)
q = [0., 0.]
A = sparse([0. 0.
-2. 0.;
0. -1.])
b = [ 1.
-2.;
-2.]
cones = [Clarabel.SecondOrderConeT(3)]
settings = Clarabel.Settings(max_iter = 15, verbose = true)
solver = Clarabel.Solver()
Clarabel.setup!(solver, P, q, A, b, cones, settings)
result = Clarabel.solve!(solver)-------------------------------------------------------------
Clarabel.jl v0.5.1 - Clever Acronym
(c) Paul Goulart
University of Oxford, 2022
-------------------------------------------------------------
problem:
variables = 2
constraints = 3
nnz(P) = 1
nnz(A) = 2
cones (total) = 1
: SecondOrder = 1, numel = 3
settings:
linear algebra: direct / qdldl, precision: Float64
max iter = 15, time limit = Inf, max step = 0.990
tol_feas = 1.0e-08, tol_gap_abs = 1.0e-08, tol_gap_rel = 1.0e-08,
static reg : on, ϵ1 = 1.0e-08, ϵ2 = 4.9e-32
dynamic reg: on, ϵ = 1.0e-13, δ = 2.0e-07
iter refine: on, reltol = 1.0e-13, abstol = 1.0e-12,
max iter = 10, stop ratio = 5.0
equilibrate: on, min_scale = 1.0e-04, max_scale = 1.0e+04
max iter = 10
iter pcost dcost gap pres dres k/t μ step
---------------------------------------------------------------------------------------------
0 4.9292e-01 -1.2888e-01 6.22e-01 1.79e-01 6.48e-17 1.00e+00 2.35e+00 ------
1 9.3926e-01 7.9266e-01 1.47e-01 6.38e-03 5.39e-17 4.74e-02 1.60e-01 9.67e-01
2 9.9952e-01 9.9654e-01 2.97e-03 6.30e-05 5.24e-20 8.14e-04 2.68e-03 9.90e-01
3 1.0000e+00 9.9997e-01 2.98e-05 6.30e-07 1.15e-21 8.16e-06 2.69e-05 9.90e-01
4 1.0000e+00 1.0000e+00 2.98e-07 6.30e-09 5.33e-17 8.16e-08 2.69e-07 9.90e-01
5 1.0000e+00 1.0000e+00 2.98e-09 6.30e-11 5.33e-17 8.16e-10 2.69e-09 9.90e-01
---------------------------------------------------------------------------------------------
Terminated with status = solved
solve time = 798μs>>> Clarabel - Results
Status: SOLVED
Iterations: 5
Objective: 1.000
Solve time: 798μs
result.x2-element Vector{Float64}:
1.0000000000000002
0.9999999997592767Using JuMP
We can solve the same problem using Clarabel.jl as the backend solver within JuMP. Here is the same problem again:
using Clarabel, JuMP
model = JuMP.Model(Clarabel.Optimizer)
set_optimizer_attribute(model, "verbose", true)
@variable(model, x[1:2])
@constraint(model, [1, 2x[1]-2, x[2] - 2] in SecondOrderCone())
@objective(model, Min, x[2]^2 )
optimize!(model)-------------------------------------------------------------
Clarabel.jl v0.5.1 - Clever Acronym
(c) Paul Goulart
University of Oxford, 2022
-------------------------------------------------------------
problem:
variables = 2
constraints = 3
nnz(P) = 1
nnz(A) = 2
cones (total) = 1
: SecondOrder = 1, numel = 3
settings:
linear algebra: direct / qdldl, precision: Float64
max iter = 200, time limit = Inf, max step = 0.990
tol_feas = 1.0e-08, tol_gap_abs = 1.0e-08, tol_gap_rel = 1.0e-08,
static reg : on, ϵ1 = 1.0e-08, ϵ2 = 4.9e-32
dynamic reg: on, ϵ = 1.0e-13, δ = 2.0e-07
iter refine: on, reltol = 1.0e-13, abstol = 1.0e-12,
max iter = 10, stop ratio = 5.0
equilibrate: on, min_scale = 1.0e-04, max_scale = 1.0e+04
max iter = 10
iter pcost dcost gap pres dres k/t μ step
---------------------------------------------------------------------------------------------
0 4.9292e-01 -1.2888e-01 6.22e-01 1.79e-01 6.48e-17 1.00e+00 2.35e+00 ------
1 9.3926e-01 7.9266e-01 1.47e-01 6.38e-03 5.39e-17 4.74e-02 1.60e-01 9.67e-01
2 9.9952e-01 9.9654e-01 2.97e-03 6.30e-05 5.24e-20 8.14e-04 2.68e-03 9.90e-01
3 1.0000e+00 9.9997e-01 2.98e-05 6.30e-07 1.15e-21 8.16e-06 2.69e-05 9.90e-01
4 1.0000e+00 1.0000e+00 2.98e-07 6.30e-09 5.33e-17 8.16e-08 2.69e-07 9.90e-01
5 1.0000e+00 1.0000e+00 2.98e-09 6.30e-11 5.33e-17 8.16e-10 2.69e-09 9.90e-01
---------------------------------------------------------------------------------------------
Terminated with status = solved
solve time = 848μsHere is the solution
JuMP.value.(x)2-element Vector{Float64}:
1.0000000000000002
0.9999999997592767and the solver termination status again
JuMP.termination_status(model)OPTIMAL::TerminationStatusCode = 1Using Convex.jl
One more time using Clarabel.jl as the backend solver within Convex.jl:
using Clarabel, Convex
x = Variable(2)
problem = minimize(square(x[2]))
problem.constraints = [norm([2x[1];x[2]] - [2;2], 2) <= 1]
solve!(problem, Clarabel.Optimizer; silent_solver = false)-------------------------------------------------------------
Clarabel.jl v0.5.1 - Clever Acronym
(c) Paul Goulart
University of Oxford, 2022
-------------------------------------------------------------
problem:
variables = 5
constraints = 9
nnz(P) = 0
nnz(A) = 9
cones (total) = 4
: Zero = 1, numel = 1
: Nonnegative = 1, numel = 2
: SecondOrder = 2, numel = (3,3)
settings:
linear algebra: direct / qdldl, precision: Float64
max iter = 200, time limit = Inf, max step = 0.990
tol_feas = 1.0e-08, tol_gap_abs = 1.0e-08, tol_gap_rel = 1.0e-08,
static reg : on, ϵ1 = 1.0e-08, ϵ2 = 4.9e-32
dynamic reg: on, ϵ = 1.0e-13, δ = 2.0e-07
iter refine: on, reltol = 1.0e-13, abstol = 1.0e-12,
max iter = 10, stop ratio = 5.0
equilibrate: on, min_scale = 1.0e-04, max_scale = 1.0e+04
max iter = 10
iter pcost dcost gap pres dres k/t μ step
---------------------------------------------------------------------------------------------
0 0.0000e+00 -2.9024e+00 2.90e+00 5.63e-01 1.51e-01 1.00e+00 2.51e+00 ------
1 5.1761e-01 -8.7840e-02 6.05e-01 2.01e-01 4.25e-02 4.40e-01 5.59e-01 8.09e-01
2 6.6756e-01 6.2124e-01 4.63e-02 2.33e-02 4.70e-03 8.74e-02 6.47e-02 9.04e-01
3 8.8598e-01 8.9192e-01 5.94e-03 9.08e-03 1.51e-03 6.05e-02 1.81e-02 8.68e-01
4 9.9565e-01 9.9523e-01 4.27e-04 2.87e-04 4.94e-05 1.35e-03 6.09e-04 9.68e-01
5 9.9983e-01 9.9982e-01 4.49e-06 1.67e-05 2.92e-06 9.86e-05 3.65e-05 9.90e-01
6 1.0000e+00 1.0000e+00 3.94e-08 2.29e-07 4.02e-08 1.38e-06 5.01e-07 9.86e-01
7 1.0000e+00 1.0000e+00 1.22e-09 7.91e-09 1.39e-09 4.77e-08 1.73e-08 9.66e-01
---------------------------------------------------------------------------------------------
Terminated with status = solved
solve time = 1.20msHere is our solution
evaluate(x)2-element Vector{Float64}:
1.0
0.9999999565274064and the solver termination status again
problem.statusOPTIMAL::TerminationStatusCode = 1This page was generated using Literate.jl.