# 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.x
2-element Vector{Float64}:
1.0000000000000002
0.9999999997592767

## Using 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 =  848μs

Here is the solution

JuMP.value.(x)
2-element Vector{Float64}:
1.0000000000000002
0.9999999997592767

and the solver termination status again

JuMP.termination_status(model)
OPTIMAL::TerminationStatusCode = 1

## Using 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 = 1.20ms

Here is our solution

evaluate(x)
2-element Vector{Float64}:
1.0
0.9999999565274064

and the solver termination status again

problem.status
OPTIMAL::TerminationStatusCode = 1