min_distortion.jl

#  Copyright 2017, Iain Dunning, Joey Huchette, Miles Lubin, and contributors
#  This Source Code Form is subject to the terms of the Mozilla Public
#  License, v. 2.0. If a copy of the MPL was not distributed with this
#  file, You can obtain one at http://mozilla.org/MPL/2.0/.
#############################################################################
# JuMP
# An algebraic modeling language for Julia
# See http://github.com/jump-dev/JuMP.jl
#############################################################################

using JuMP, SCS, Test

"""
    example_min_distortion()

This example arises from computational geometry, in particular the problem of
embedding a general finite metric space into a euclidean space.

It is known that the 4-point metric space defined by the star graph:
  x
   \\
    x — x
   /
  x
where distances are computed by length of the shortest path between vertices,
cannot be exactly embedded into a euclidean space of any dimension.

Here we will formulate and solve an SDP to compute the best possible embedding,
that is, the embedding f() that minimizes the distortion c such that
    (1 / c) * D(a, b) ≤ ||f(a) - f(b)|| ≤ D(a, b)
for all points (a, b), where D(a, b) is the distance in the metric space.

Any embedding can be characterized by its Gram matrix Q, which is PSD, and
    ||f(a) - f(b)||^2 = Q[a, a] + Q[b, b] - 2 * Q[a, b]
We can therefore constrain
    D[i, j]^2 ≤ Q[i, i] + Q[j, j] - 2 * Q[i, j] ≤ c^2 * D[i, j]^2
and minimize c^2, which gives us the SDP formulation below.

For more detail, see "Lectures on discrete geometry" by J. Matoušek, Springer,
2002, pp. 378-379.
"""
function example_min_distortion()
    model = Model(SCS.Optimizer)
    set_silent(model)
    D = [0.0 1.0 1.0 1.0;
         1.0 0.0 2.0 2.0;
         1.0 2.0 0.0 2.0;
         1.0 2.0 2.0 0.0]
    @variable(model, c² >= 1.0)
    @variable(model, Q[1:4, 1:4], PSD)
    for i in 1:4
        for j in (i + 1):4
            @constraint(model, D[i, j]^2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j])
            @constraint(model, Q[i, i] + Q[j, j] - 2 * Q[i, j] <= c² * D[i, j]^2)
        end
    end
    @objective(model, Min, c²)
    JuMP.optimize!(model)

    @test JuMP.termination_status(model) == MOI.OPTIMAL
    @test JuMP.primal_status(model) == MOI.FEASIBLE_POINT
    @test JuMP.objective_value(model) ≈ 4/3 atol = 1e-4
end

example_min_distortion()