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) 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 ()