Chebyshev center
Boyd & Vandenberghe, "Convex Optimization" Joëlle Skaf - 08/16/05
Adapted for Convex.jl by Karanveer Mohan and David Zeng - 26/05/14
The goal is to find the largest Euclidean ball (that is, its center and radius) that lies in a polyhedron described by affine inequalities in this fashion: $P = \{x : a_i'*x \leq b_i, i=1,\ldots,m \}$ where $x \in \mathbb{R}^2$.
using Convex
using LinearAlgebra
import SCS
Generate the input data
a1 = [2; 1];
a2 = [2; -1];
a3 = [-1; 2];
a4 = [-1; -2];
b = ones(4, 1);
Create and solve the model
r = Variable(1)
x_c = Variable(2)
constraints = [
a1' * x_c + r * norm(a1, 2) <= b[1],
a2' * x_c + r * norm(a2, 2) <= b[2],
a3' * x_c + r * norm(a3, 2) <= b[3],
a4' * x_c + r * norm(a4, 2) <= b[4],
]
p = maximize(r, constraints)
solve!(p, SCS.Optimizer; silent = true)
Problem statistics
problem is DCP : true
number of variables : 2 (3 scalar elements)
number of constraints : 4 (4 scalar elements)
number of coefficients : 16
number of atoms : 12
Solution summary
termination status : OPTIMAL
primal status : FEASIBLE_POINT
dual status : FEASIBLE_POINT
objective value : 0.4472
Expression graph
maximize
└─ real variable (id: 139…148)
subject to
├─ ≤ constraint (affine)
│ └─ + (affine; real)
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ [-1.0;;]
├─ ≤ constraint (affine)
│ └─ + (affine; real)
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ [-1.0;;]
├─ ≤ constraint (affine)
│ └─ + (affine; real)
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ ├─ * (affine; real)
│ │ ├─ …
│ │ └─ …
│ └─ [-1.0;;]
⋮
Generate the figure
x = range(-1.5, stop = 1.5, length = 100);
theta = 0:pi/100:2*pi;
using Plots
plot(x, x -> -x * a1[1] / a1[2] + b[1] / a1[2])
plot!(x, x -> -x * a2[1] / a2[2] + b[2] / a2[2])
plot!(x, x -> -x * a3[1] / a3[2] + b[3] / a3[2])
plot!(x, x -> -x * a4[1] / a4[2] + b[4] / a4[2])
plot!(
evaluate(x_c)[1] .+ evaluate(r) * cos.(theta),
evaluate(x_c)[2] .+ evaluate(r) * sin.(theta),
linewidth = 2,
)
plot!(
title = "Largest Euclidean ball lying in a 2D polyhedron",
legend = nothing,
)
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