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Find largest-area inscribed polygon Description Given n, find the n-sided polygon of largest area inscribed in the unit circle. Further explanation of this example:
'Xpress Python Reference Manual'
Source Files polygon.py 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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