import seaborn as sns

import matplotlib.pyplot as plt
from matplotlib import cm
from mpl_toolkits.mplot3d import Axes3D
from numpy import *

fig = plt.figure()
ax = fig.gca(projection='3d')

x = np.linspace(-7, 7, 100)
y = np.linspace(-7, 7, 100)

X, Y = np.meshgrid(x, y)
Z = 10*(X**2 - Y**2)**2 - (1 - X)**2 + X*Y

ax.plot_surface(X, Y, Z, rstride=8, cstride=8, alpha=0.3)

fig = plt.figure()

ax = fig.gca(projection='3d')

mult = 20

x = np.linspace(-7, 7, 100)
y = np.linspace(-7, 7, 100)

X, Y = np.meshgrid(x, y)
Z = 10*(X**2 - Y**2)**2 - (1 - X)**2 + X*Y

cset = ax.contour(X, Y, Z, [1, 4, 9, 16], cmap=cm.coolwarm)

ax.clabel(cset, fontsize=90, inline=1)


fig = plt.figure()
ax = fig.gca(projection='3d')

X, Y = numpy.linspace(left, right, 200), numpy.linspace(left, right, 200)
(X, Y), Z = numpy.meshgrid(X, Y), f(X, Y)

ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm, linewidth=0, antialiased=False)

from scipy.optimize import minimize
rnd = lambda: 10*numpy.random.random() - 5
g = lambda (x, y): f(x, y)

n_sample = 10000

minimum = np.array(
    [minimize(g, x0=(rnd(), rnd()), bounds=((-5,5), (-5, 5)), method='SLSQP')['x'] 
     for _ in range(n_sample)]
)

maximum = np.array(
    [minimize(lambda x: -g(x), x0=(rnd(), rnd()), bounds=((-5,5), (-5, 5)), method='SLSQP')['x'] 
     for _ in range(n_sample)]
)

fig = plt.figure()
ax = fig.gca(projection='3d')

ax.scatter(minimum[:, 0], minimum[:, 1], g((minimum[:, 0], minimum[:, 1])), c='r', alpha=0.5)
ax.scatter(maximum[:, 0], maximum[:, 1], g((maximum[:, 0], maximum[:, 1])), c='b', alpha=0.5)

g_value_in_loc_min = g((minimum[:, 0], minimum[:, 1]))
x_min = np.min(g_value_in_loc_min)
print x_min

def f1(x1, x2): return x1**2 + x2**2 - 0.5*x1*x2
def g(x1, x2): return x2**2 - x1**2 - 3

def u(x1): return 2*x1 - 0.5*x1*numpy.sqrt(x1**2 + 3) + 3


fig = plt.figure()
ax = fig.gca(projection='3d')

X, Y = numpy.linspace(left, right, 200), numpy.linspace(left, right, 200)
(X, Y), Z = numpy.meshgrid(X, Y), f1(X, Y)

ax.plot_surface(X, Y, Z, color='b')

X, Y = numpy.linspace(left, right, 100), numpy.linspace(left, right, 100)
(X, Y), Z = numpy.meshgrid(X, Y), g(X, Y)

ax.plot_surface(X, Y, Z, color='r')


Y = numpy.linspace(-10, 10, 1000)
pylab.plot(Y, u(Y))

y = Out[46]['x'][0]
x = y**2+3
print(x, y, f(x, y))