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: 811…477)
  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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