import xpress as xp
import math
import matplotlib.pyplot as plt
from matplotlib.path import Path
import matplotlib.patches as patches
# Problem: given n, find the n-sided polygon of largest area inscribed
# in the unit circle.
#
# While it is natural to prove that all vertices of a global optimum
# reside on the unit circle, here we formulate the problem so that
# every vertex i is at distance rho[i] from the center and at angle
# theta[i]. We would certainly expect that the local optimum found has
# all rho's are equal to 1.
N = 9
Vertices = range (N)
# Declare variables
rho = [xp.var (name = 'rho_{}' .format (i), lb = 1e-5, ub = 1.0) for i in Vertices]
theta = [xp.var (name = 'theta_{}'.format (i), lb = -math.pi, ub = math.pi) for i in Vertices]
p = xp.problem ()
p.addVariable (rho, theta)
# The objective function is the total area of the polygon. Considering
# the segment S[i] joining the center to the i-th vertex and A(i,j)
# the area of the triangle defined by the two segments S[i] and S[j],
# the objective function is
#
# A[0,1] + A[1,2] + ... + A[N-1,0]
#
# Where A[i,i+1] is given by
#
# 1/2 * rho[i] * rho[i+1] * sin (theta[i+1] - theta[i])
p.setObjective (0.5 * (xp.Sum (rho[i] * rho[i-1] * xp.sin (theta[i] - theta[i-1]) for i in Vertices if i != 0) # sum of the first N-1 triangle areas
+ rho[0] * rho[N-1] * xp.sin (theta[0] - theta[N-1])), sense = xp.maximize) # plus area between segments N and 1
# Angles are in increasing order, and should be different (the solver
# finds a bad local optimum otherwise
p.addConstraint (theta[i] >= theta[i-1] + 1e-4 for i in Vertices if i != 0)
# solve the problem
p.solve ()
# The following command saves the final problem onto a file
#
# p.write ('polygon{}'.format (N), 'lp')
rho_sol = p.getSolution (rho)
theta_sol = p.getSolution (theta)
x_coord = [rho_sol[i] * math.cos (theta_sol[i]) for i in Vertices]
y_coord = [rho_sol[i] * math.sin (theta_sol[i]) for i in Vertices]
vertices = [(x_coord [i], y_coord [i]) for i in Vertices] + [(x_coord [0], y_coord [0])]
moves = [Path.MOVETO] + [Path.LINETO] * (N-1) + [Path.CLOSEPOLY]
path = Path (vertices, moves)
fig = plt.figure ()
sp = fig.add_subplot (111)
patch = patches.PathPatch (path, lw=1)
sp.add_patch (patch)
# Define bounds of picture, as it would be [0,1]^2 otherwise
sp.set_xlim (-1.1, 1.1)
sp.set_ylim (-1.1, 1.1)
plt.show ()
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