Basic 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
To solve the problem directly within Clarabel.jl, start by creating the solver and settings:
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 v0.5.0 - 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 = 688μs>>> Clarabel - Results
Status: SOLVED
Iterations: 5
Objective: 1.000
Solve time: 688μs
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 v0.5.0 - 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 = 707μsHere 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 v0.5.0 - 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 = 959μsHere 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.