File:Nelder-Mead Simionescu.gif
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Summary
DescriptionNelder-Mead Simionescu.gif |
English: Animated Nelder-Mead minimum search of Simionescu's function. |
Date | |
Source | Own work |
Author | nicoguaro |
GIF development InfoField | This plot was created with Matplotlib. |
Source code InfoField | Python codefrom 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()
|
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
Items portrayed in this file
depicts
26 June 2018
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 21:47, 26 June 2018 | 1,000 × 1,000 (1.31 MB) | Nicoguaro | Improve line widths |
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