Example: ellipsoid approximation

This tutorial was generated using Literate.jl. Download the source as a .jl file.

This tutorial considers the problem of computing extremal ellipsoids: finding ellipsoids that best approximate a given set. As an extension, we show how to use JuMP to inspect the bridges that were used, and how to explore alternative formulations.

The model comes from Section 4.9 of (Ben-Tal and Nemirovski, 2001).

For a related example, see also the Example: minimal ellipses tutorial.

Required packages

This tutorial uses the following packages:

using JuMP
import Clarabel
import LinearAlgebra
import Plots
import Random
import Test

Problem formulation

Suppose that we are given a set $\mathcal{S}$ consisting of $m$ points in $n$-dimensional space:

\[\mathcal{S} = \{ x_1, \ldots, x_m \} \subset \mathbb{R}^n\]

Our goal is to determine an optimal vector $c \in \mathbb{R}^n$ and an optimal $n \times n$ real symmetric matrix $D$ such that the ellipse:

\[E(D, c) = \{ x : (x - c)^\top D ( x - c) \leq 1 \},\]

contains $\mathcal{S}$ and has the smallest possible volume.

The optimal $D$ and $c$ are given by the optimization problem:

\[\begin{aligned} \max \quad & t \\ \text{s.t.} \quad & Z \succeq 0 \\ & \begin{bmatrix} s & z^\top \\ z & Z \\ \end{bmatrix} \succeq 0 \\ & x_i^\top Z x_i - 2x_i^\top z + s \leq 1 & \quad i=1, \ldots, m \\ & t \le \sqrt[n]{\det(Z)}, \end{aligned}\]

where $D = Z_*$ and $c = Z_*^{-1} z_*$.

Data

We first need to generate some points to work with.

function generate_point_cloud(
    m;            # number of 2-dimensional points
    a = 10,       # scaling in x direction
    b = 2,        # scaling in y direction
    rho = π / 6,  # rotation of points around origin
    random_seed = 1,
)
    rng = Random.MersenneTwister(random_seed)
    P = randn(rng, Float64, m, 2)
    Phi = [a*cos(rho) a*sin(rho); -b*sin(rho) b*cos(rho)]
    S = P * Phi
    return S
end

generate_point_cloud (generic function with 1 method)

For the sake of this example, let's take $m = 100$:

S = generate_point_cloud(100);

We will visualise the points (and ellipse) using the Plots package:

r = 1.1 * maximum(abs.(S))
plot = Plots.scatter(
    S[:, 1],
    S[:, 2];
    xlim = (-r, r),
    ylim = (-r, r),
    label = nothing,
    c = :green,
    shape = :x,
    size = (600, 600),
)

JuMP formulation

Now let's build and the JuMP model. We'll compute $D$ and $c$ after the solve.

model = Model(Clarabel.Optimizer)
set_silent(model)
m, n = size(S)
@variable(model, z[1:n])
@variable(model, Z[1:n, 1:n], PSD)
@variable(model, s)
@variable(model, t)
@constraint(model, [s z'; z Z] >= 0, PSDCone())
@constraint(
    model,
    [i in 1:m],
    S[i, :]' * Z * S[i, :] - 2 * S[i, :]' * z + s <= 1,
)
@constraint(model, [t; vec(Z)] in MOI.RootDetConeSquare(n))
@objective(model, Max, t)
optimize!(model)
assert_is_solved_and_feasible(model)
solution_summary(model)

solution_summary(; result = 1, verbose = false)
├ solver_name          : Clarabel
├ Termination
│ ├ termination_status : OPTIMAL
│ ├ result_count       : 1
│ └ raw_status         : SOLVED
├ Solution (result = 1)
│ ├ primal_status        : FEASIBLE_POINT
│ ├ dual_status          : FEASIBLE_POINT
│ ├ objective_value      : 7.92350e-03
│ └ dual_objective_value : 7.92350e-03
└ Work counters
  ├ solve_time (sec)   : 1.69295e-02
  └ barrier_iterations : 16

Results

After solving the model to optimality we can recover the solution in terms of $D$ and $c$:

D = value.(Z)

2×2 Matrix{Float64}:
  0.012616  -0.02132
 -0.02132    0.0410053

c = D \ value.(z)

2-element Vector{Float64}:
 -1.6318778797029514
 -0.670458237995941

We can check that each point lies inside the ellipsoid, by checking if the largest normalized radius is less than 1:

largest_radius = maximum(map(x -> (x - c)' * D * (x - c), eachrow(S)))

0.9999999991403553

Finally, overlaying the solution in the plot we see the minimal volume approximating ellipsoid:

P = sqrt(D)
q = -P * c
data = [tuple(P \ [cos(θ) - q[1], sin(θ) - q[2]]...) for θ in 0:0.05:(2pi+0.05)]
Plots.plot!(plot, data; c = :crimson, label = nothing)

Alternative formulations

The formulation of model uses MOI.RootDetConeSquare. However, because Clarabel does not natively support this cone, JuMP automatically reformulates the problem into an equivalent problem that Clarabel does support. You can see the reformulation that JuMP chose using print_active_bridges:

print_active_bridges(model)

 * Unsupported objective: MOI.VariableIndex
 |  bridged by:
 |   MOIB.Objective.FunctionConversionBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.VariableIndex}
 |  may introduce:
 |   * Supported objective: MOI.ScalarAffineFunction{Float64}
 * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.LessThan{Float64}
 |  bridged by:
 |   MOIB.Constraint.LessToGreaterBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}}
 |  may introduce:
 |   * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.GreaterThan{Float64}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Nonnegatives, MOI.ScalarAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeSquare
 |  bridged by:
 |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, MOI.PositiveSemidefiniteConeSquare}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |  bridged by:
 |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 |   * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros
 * Unsupported constraint: MOI.VectorOfVariables-in-MOI.PositiveSemidefiniteConeTriangle
 |  bridged by:
 |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables}
 |  may introduce:
 |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 * Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeSquare
 |  bridged by:
 |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorOfVariables, MOI.ScalarAffineFunction{Float64}, MOI.RootDetConeTriangle, MOI.RootDetConeSquare}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeTriangle
 |   |  bridged by:
 |   |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.VectorOfVariables}
 |   |  may introduce:
 |   |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |   |  bridged by:
 |   |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |   |  may introduce:
 |   |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 |   |   * Unsupported constraint: MOI.VectorOfVariables-in-MOI.GeometricMeanCone
 |   |   |  bridged by:
 |   |   |   MOIB.Constraint.GeoMeanToPowerBridge{Float64, MOI.VectorOfVariables}
 |   |   |  may introduce:
 |   |   |   * Supported constraint: MOI.VectorOfVariables-in-MOI.PowerCone{Float64}
 |   |   |   * Supported variable: MOI.Nonnegatives
 |   |   * Supported variable: MOI.Reals
 |   * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros

There's a lot going on here, but the first bullet is:

* Unsupported objective: MOI.VariableIndex
|  bridged by:
|   MOIB.Objective.FunctionConversionBridge{Float64}
|  may introduce:
|   * Supported objective: MOI.ScalarAffineFunction{Float64}

This says that Clarabel does not support a MOI.VariableIndex objective function, and that JuMP used a MOI.Bridges.Objective.FunctionConversionBridge to convert it into a MOI.ScalarAffineFunction{Float64} objective function.

We can leave JuMP to do the reformulation, or we can rewrite our model to have an objective function that Clarabel natively supports:

@objective(model, Max, 1.0 * t + 0.0);

Re-printing the active bridges:

print_active_bridges(model)

 * Supported objective: MOI.ScalarAffineFunction{Float64}
 * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.LessThan{Float64}
 |  bridged by:
 |   MOIB.Constraint.LessToGreaterBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}}
 |  may introduce:
 |   * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.GreaterThan{Float64}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Nonnegatives, MOI.ScalarAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeSquare
 |  bridged by:
 |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.ScalarAffineFunction{Float64}, MOI.PositiveSemidefiniteConeTriangle, MOI.PositiveSemidefiniteConeSquare}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |  bridged by:
 |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 |   * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros
 * Unsupported constraint: MOI.VectorOfVariables-in-MOI.PositiveSemidefiniteConeTriangle
 |  bridged by:
 |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables}
 |  may introduce:
 |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 * Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeSquare
 |  bridged by:
 |   MOIB.Constraint.SquareBridge{Float64, MOI.VectorOfVariables, MOI.ScalarAffineFunction{Float64}, MOI.RootDetConeTriangle, MOI.RootDetConeSquare}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorOfVariables-in-MOI.RootDetConeTriangle
 |   |  bridged by:
 |   |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorOfVariables, MOI.VectorOfVariables}
 |   |  may introduce:
 |   |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |   |  bridged by:
 |   |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |   |  may introduce:
 |   |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 |   |   * Unsupported constraint: MOI.VectorOfVariables-in-MOI.GeometricMeanCone
 |   |   |  bridged by:
 |   |   |   MOIB.Constraint.GeoMeanToPowerBridge{Float64, MOI.VectorOfVariables}
 |   |   |  may introduce:
 |   |   |   * Supported constraint: MOI.VectorOfVariables-in-MOI.PowerCone{Float64}
 |   |   |   * Supported variable: MOI.Nonnegatives
 |   |   * Supported variable: MOI.Reals
 |   * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.EqualTo{Float64}
 |   |  bridged by:
 |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Zeros, MOI.ScalarAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Zeros

we get * Supported objective: MOI.ScalarAffineFunction{Float64}.

We can manually implement some other reformulations to change our model to something that Clarabel more closely supports by:

Replacing the MOI.VectorOfVariables in MOI.PositiveSemidefiniteConeTriangle constraint @variable(model, Z[1:n, 1:n], PSD) with the MOI.VectorAffineFunction in MOI.PositiveSemidefiniteConeTriangle @constraint(model, Z >= 0, PSDCone()).
Replacing the MOI.VectorOfVariables in MOI.PositiveSemidefiniteConeSquare constraint [s z'; z Z] >= 0, PSDCone() with the MOI.VectorAffineFunction in MOI.PositiveSemidefiniteConeTriangle @constraint(model, LinearAlgebra.Symmetric([s z'; z Z]) >= 0, PSDCone()).
Replacing the MOI.ScalarAffineFunction in MOI.GreaterThan constraints with the vectorized equivalent of MOI.VectorAffineFunction in MOI.Nonnegatives
Replacing the MOI.VectorOfVariables in MOI.RootDetConeSquare constraint with MOI.VectorAffineFunction in MOI.RootDetConeTriangle.

model = Model(Clarabel.Optimizer)
set_silent(model)
@variable(model, z[1:n])
@variable(model, s)
@variable(model, t)
# The former @variable(model, Z[1:n, 1:n], PSD)
@variable(model, Z[1:n, 1:n], Symmetric)
@constraint(model, Z >= 0, PSDCone())
# The former [s z'; z Z] >= 0, PSDCone()
@constraint(model, LinearAlgebra.Symmetric([s z'; z Z]) >= 0, PSDCone())
# The former constraint S[i, :]' * Z * S[i, :] - 2 * S[i, :]' * z + s <= 1
f = [1 - S[i, :]' * Z * S[i, :] + 2 * S[i, :]' * z - s for i in 1:m]
@constraint(model, f in MOI.Nonnegatives(m))
# The former constraint [t; vec(Z)] in MOI.RootDetConeSquare(n)
@constraint(model, 1 * [t; triangle_vec(Z)] .+ 0 in MOI.RootDetConeTriangle(n))
# The former @objective(model, Max, t)
@objective(model, Max, 1 * t + 0)
optimize!(model)
assert_is_solved_and_feasible(model)
solve_time_1 = solve_time(model)

0.002221629

This formulation gives the much smaller graph:

print_active_bridges(model)

 * Supported objective: MOI.ScalarAffineFunction{Float64}
 * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |  bridged by:
 |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |  may introduce:
 |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.RootDetConeTriangle
 |  bridged by:
 |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |  bridged by:
 |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.GeometricMeanCone
 |   |  bridged by:
 |   |   MOIB.Constraint.GeoMeanToPowerBridge{Float64, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PowerCone{Float64}
 |   |   * Supported variable: MOI.Nonnegatives
 |   * Supported variable: MOI.Reals

Note that we still need to bridge MOI.PositiveSemidefiniteConeTriangle constraints because Clarabel uses the MOI.Scaled PSD cone.

model = Model(Clarabel.Optimizer)
set_silent(model)
@variable(model, z[1:n])
@variable(model, s)
@variable(model, t)
@variable(model, Z[1:n, 1:n], Symmetric)
# The former @constraint(model, Z in PSDCone())
f = triangle_vec(Z)
scale_f = [1.0, sqrt(2), 1.0]
@constraint(
    model,
    scale_f .* f in MOI.Scaled(MOI.PositiveSemidefiniteConeTriangle(n)),
)
# The former LinearAlgebra.Symmetric([s z'; z Z]) >= 0, PSDCone()
g = triangle_vec(LinearAlgebra.Symmetric([s z'; z Z]))
scale_g = [1.0, sqrt(2), 1.0, sqrt(2), sqrt(2), 1.0]
@constraint(
    model,
    scale_g .* g in MOI.Scaled(MOI.PositiveSemidefiniteConeTriangle(1 + n)),
)
f = [1 - S[i, :]' * Z * S[i, :] + 2 * S[i, :]' * z - s for i in 1:m]
@constraint(model, f in MOI.Nonnegatives(m))
@constraint(model, 1 * [t; triangle_vec(Z)] .+ 0 in MOI.RootDetConeTriangle(n))
@objective(model, Max, 1 * t + 0)
optimize!(model)
assert_is_solved_and_feasible(model)
solve_time_2 = solve_time(model)

0.0021437820000000003

This formulation gives the much smaller graph:

print_active_bridges(model)

 * Supported objective: MOI.ScalarAffineFunction{Float64}
 * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.RootDetConeTriangle
 |  bridged by:
 |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |  bridged by:
 |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.GeometricMeanCone
 |   |  bridged by:
 |   |   MOIB.Constraint.GeoMeanToPowerBridge{Float64, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PowerCone{Float64}
 |   |   * Supported variable: MOI.Nonnegatives
 |   * Supported variable: MOI.Reals
 * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}

Now there is only a single Unsupported constraint bullet, showing how JuMP reformulated the MOI.RootDetConeTriangle constraint by adding a mix of MOI.PositiveSemidefiniteConeTriangle and MOI.GeometricMeanCone constraints.

Because Clarabel doesn't natively support the MOI.GeometricMeanCone, these constraints were further bridged using a MOI.Bridges.Constraint.GeoMeanToPowerBridge to a series of MOI.PowerCone constraints.

However, there are many other ways that a MOI.GeometricMeanCone can be reformulated into something that Clarabel supports. Let's see what happens if we use remove_bridge to remove the MOI.Bridges.Constraint.GeoMeanToPowerBridge:

remove_bridge(model, MOI.Bridges.Constraint.GeoMeanToPowerBridge)
optimize!(model)
assert_is_solved_and_feasible(model)

This time, the solve took:

solve_time_3 = solve_time(model)

0.00213194

where previously it took

solve_time_2

0.0021437820000000003

Why was the solve time different?

print_active_bridges(model)

 * Supported objective: MOI.ScalarAffineFunction{Float64}
 * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives
 * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.RootDetConeTriangle
 |  bridged by:
 |   MOIB.Constraint.RootDetBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |  may introduce:
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.PositiveSemidefiniteConeTriangle
 |   |  bridged by:
 |   |   MOIB.Constraint.SetDotScalingBridge{Float64, MOI.PositiveSemidefiniteConeTriangle, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}
 |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.GeometricMeanCone
 |   |  bridged by:
 |   |   MOIB.Constraint.GeoMeantoRelEntrBridge{Float64, MOI.VectorOfVariables, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |  may introduce:
 |   |   * Unsupported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.RelativeEntropyCone
 |   |   |  bridged by:
 |   |   |   MOIB.Constraint.RelativeEntropyBridge{Float64, MOI.ScalarAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}, MOI.VectorAffineFunction{Float64}}
 |   |   |  may introduce:
 |   |   |   * Unsupported constraint: MOI.ScalarAffineFunction{Float64}-in-MOI.GreaterThan{Float64}
 |   |   |   |  bridged by:
 |   |   |   |   MOIB.Constraint.VectorizeBridge{Float64, MOI.VectorAffineFunction{Float64}, MOI.Nonnegatives, MOI.ScalarAffineFunction{Float64}}
 |   |   |   |  may introduce:
 |   |   |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Nonnegatives
 |   |   |   * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.ExponentialCone
 |   |   |   * Supported variable: MOI.Reals
 |   |   * Supported variable: MOI.Nonnegatives
 |   * Supported variable: MOI.Reals
 * Supported constraint: MOI.VectorAffineFunction{Float64}-in-MOI.Scaled{MOI.PositiveSemidefiniteConeTriangle}

This time, JuMP used a MOI.Bridges.Constraint.GeoMeantoRelEntrBridge to reformulate the constraint into a set of MOI.RelativeEntropyCone constraints, which were further reformulated into a set of supported MOI.ExponentialCone constraints.

Since the two models are equivalent, we can conclude that for this particular model, the formulations have similar performance.

In general though, the performance of a particular reformulation is problem- and solver-specific. Therefore, JuMP chooses to minimize the number of bridges in the default reformulation, leaving you to explore alternative formulations using the tools and techniques shown in this tutorial.