from numpy import cos, arctan2 import numpy as np import matplotlib.pyplot as plt import matplotlib.animation as animation  plt.rcParams["font.size"] = 10 plt.rcParams["mathtext.fontset"] = "cm"  # Minimization def nelder_mead_step(fun, verts, alpha=1, gamma=2, rho=0.5,                      sigma=0.5, beta=1.0):     """Nelder-Mead iteration according to Wikipedia _[1]               References     ----------      .. [1] Wikipedia contributors. "Nelder–Mead method." Wikipedia,          The Free Encyclopedia. Wikipedia, The Free Encyclopedia,          1 Sep. 2016. Web. 20 Sep. 2016.      """     nverts, _ = verts.shape     f = np.apply_along_axis(fun, 1, verts, beta=beta)     # 1. Order     order = np.argsort(f)     verts = verts[order, :]     f = f[order]     # 2. Calculate xo, the centroid"     xo = verts[:-1, :].mean(axis=0)     # 3. Reflection     xr = xo + alpha*(xo - verts[-1, :])     fr = fun(xr, beta)     if f[0]<=fr and fr<f[-2]:         new_verts = np.vstack((verts[:-1, :], xr))     # 4. Expansion     elif fr<f[0]:         xe = xo + gamma*(xr - xo)         fe = fun(xe, beta)         if fe < fr:             new_verts = np.vstack((verts[:-1, :], xe))         else:             new_verts = np.vstack((verts[:-1, :], xe))     # 5. Contraction     else:         xc = xo + rho*(verts[-1, :] - xo)         fc = fun(xc, beta)         if fc < f[-1]:             new_verts = np.vstack((verts[:-1, :], xc))     # 6. Shrink         else:             new_verts = np.zeros_like(verts)             new_verts[0, :] = verts[0, :]             for k in range(1, nverts):                 new_verts[k, :] = sigma*(verts[k,:] - verts[0,:])       return new_verts  def fun(x, beta=1.0):     """Simionescu function using log-barrier method"""     x1, x2 = x     if x1**2 + x2**2 < (1 + 0.2*cos(8*arctan2(x1, x2)))**2:         barrier = -beta*np.log((1 + 0.2*cos(8*arctan2(x1, x2)))**2 - x1**2 - x2**2)     else:         barrier = np.inf     return x1*x2 + barrier  # Animation  def data_gen(num):     plt.gca().cla     x0 = np.array([0.4, -0.6])     x1 = np.array([-0.3, -0.6])     x2 = np.array([0.7, 0.6])     verts = np.vstack((x0, x1, x2))     beta = 1.0     for cont in range(num):         verts = nelder_mead_step(fun, verts, beta=beta)         beta /=2     # Plots     plt.cla()     poly = plt.Polygon(verts, facecolor="none", edgecolor="k",                        linewidth=0.5, zorder=4)     plt.gca().add_patch(poly)     x1, x2 = np.mgrid[-1.25:1.25:101j, -1.25:1.25:101j]     z = x1*x2     cons = x1**2 + x2**2 - (1 + 0.2*cos(8*arctan2(x1, x2)))**2     z[cons > 0.02] = np.nan     levels = np.linspace(-1, 1, 30)     plt.contour(x1, x2, z, levels, cmap="seismic", linewidths=1)     plt.contour(x1, x2, cons, [0], colors="black", linewidths=1)     plt.axis("image")     plt.xlabel(r"$x$", fontsize=14)     plt.ylabel(r"$y$", fontsize=14)  fig = plt.figure(figsize=(5, 5)) ani = animation.FuncAnimation(fig, data_gen, range(25)) ani.save("Nelder-Mead_Simionescu.gif", writer='imagemagick', fps=2,          dpi=200) plt.show()