Question

Try to find the global optimal point of the function (reading Python for finance 2nd edition - Chapter 11. Mathematical Tools).

def fm(p):
    x, y = p
    return (np.sin(x) + 0.05 * x ** 2
          + np.sin(y) + 0.05 * y ** 2)

scipy.optimize.basinhopping says it finds the global minimum.

Find the global minimum of a function using the basin-hopping algorithm

However, it looks it does not find the global optimal point. Why is this and how can make it find the global optimal?

import numpy as np
import scipy.optimize as sco 
from pylab import plt, mpl
from mpl_toolkits.mplot3d import Axes3D

plt.style.use('seaborn')
mpl.rcParams['font.family'] = 'serif'
%matplotlib notebook

x = np.linspace(-10, 10, 50)
y = np.linspace(-10, 10, 50)
X, Y = np.meshgrid(x, y)
Z = fm((X, Y))

optima = sco.basinhopping(
    fo,
    (-10, 10),
    stepsize=0.1
)
# Optimal 
optima['x'] = np.append(optima['x'], fm((optima['x'][0], optima['x'][1])))
optima

Result

                        fun: 3.5447966927667616
 lowest_optimization_result:       fun: 3.5447966927667616
 hess_inv: array([[9.01401735e-01, 1.68119491e-03],
       [1.68119491e-03, 2.84089686e+00]])
      jac: array([2.98023224e-08, 2.98023224e-08])
  message: 'Optimization terminated successfully.'
     nfev: 24
      nit: 5
     njev: 6
   status: 0
  success: True
        x: array([-1.42755175,  9.67888407])
                    message: ['requested number of basinhopping iterations completed successfully']
      minimization_failures: 0
                       nfev: 2516
                        nit: 100
                       njev: 629
                          x: array([-1.42755175,  9.67888407,  3.54479669])

Plot:

fig = plt.figure(figsize=(10, 6))
ax = fig.gca(projection='3d')

# Function
surf = ax.plot_surface(
    X, Y, Z, 
    rstride=2, 
    cstride=2,
    cmap='coolwarm', 
    linewidth=0.5,
    antialiased=True
)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('f(x, y)')

# Optima
ax.plot(
    [optima['x'][0]], [optima['x'][1]], [optima['x'][2]], 
    color='r', marker='X', markersize=15
)
fig.colorbar(surf, shrink=0.5, aspect=5)

Brute

If used [scipy.optimize.brute][5], it may be able to find near point.

OX = []
OY = []
output = True
def fo(p):
    x, y = p
    z = np.sin(x) + 0.05 * x ** 2 + np.sin(y) + 0.05 * y ** 2
    if output == True:
        #print('%8.4f | %8.4f | %8.4f' % (x, y, z))
        OX.append(x)
        OY.append(y)
    return z

optima = sco.brute(
    fo, 
    (
        (-10, 10.1, 2), # Step X from -10 to 10.1 by interval 2
        (-10, 10.1, 2)  # Step Y from -10 to 10.1 by interval 2
    ), 
    finish=None
)  
OZ = fm((np.array(OX), np.array(OY)))

# Optimal 
optima = np.append(optima, fm((optima[0], optima[1])))

fig = plt.figure(figsize=(10, 6))
ax = fig.gca(projection='3d')

# Function
surf = ax.plot_surface(
    X, Y, Z, 
    rstride=2, 
    cstride=2,
    cmap='coolwarm', 
    linewidth=0.5,
    antialiased=True
)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('f(x, y)')

# Brute computation tack
ax.plot(np.array(OX), np.array(OY), np.array(OZ), ls="--", color='k', linewidth=0.5)

# Optima
ax.plot(
    [optima[0]], [optima[1]], [optima[2]], 
    color='r', marker='X', markersize=15
)
fig.colorbar(surf, shrink=0.5, aspect=5)

SHGO

scipy.optimize.shgo seems to work, too.

Finds the global minimum of a function using SHG optimization.

optima = sco.shgo(
    fo,
    [(-10, 10), (-10, 10)]
)
# Optimal 
optima['x'] = np.append(optima['x'], fm((optima['x'][0], optima['x'][1])))

fig = plt.figure(figsize=(10, 6))
ax = fig.gca(projection='3d')

# Function
surf = ax.plot_surface(
    X, Y, Z, 
    rstride=2, 
    cstride=2,
    cmap='coolwarm', 
    linewidth=0.5,
    antialiased=True
)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('f(x, y)')

# Optima
ax.plot(
    [optima['x'][0]], [optima['x'][1]], [optima['x'][2]], 
    color='r', marker='X', markersize=15
)
fig.colorbar(surf, shrink=0.5, aspect=5)

Clarification

Even though it is "global Optimization", are there conditions or limitations to consider to make sure they find the global optimal?

Global Optimization
basinhopping(func, x0[, niter, T, stepsize, …])
Find the global minimum of a function using the basin-hopping algorithm
brute(func, ranges[, args, Ns, full_output, …])
Minimize a function over a given range by brute force.
differential_evolution(func, bounds[, args, …])
Finds the global minimum of a multivariate function.
shgo(func, bounds[, args, constraints, n, …])
Finds the global minimum of a function using SHG optimization.
dual_annealing(func, bounds[, args, …])
Find the global minimum of a function using Dual Annealing.

References

• Python for finance 2nd edition - Chapter 11. Mathematical Tools
• Which Optimization method to use?
• Optimization and Root Finding (scipy.optimize)