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.
using JuMP import SCS import Test 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) optimize!(model) Test.@test termination_status(model) == MOI.OPTIMAL Test.@test primal_status(model) == MOI.FEASIBLE_POINT Test.@test objective_value(model) ≈ 4/3 atol = 1e-4 return end example_min_distortion()
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