python - scipy.optimize get's trapped in local minima. What can I do?

Question

 from numpy import *; from scipy.optimize import *; from math import *
def f(X):
    x=X[0];    y=X[1]
    return x**4-3.5*x**3-2*x**2+12*x+y**2-2*y

bnds = ((1,5), (0, 2))
min_test = minimize(f,[1,0.1], bounds = bnds); 
print(min_test.x)

My function f(X)has a local minima at x=2.557, y=1 which I should be able to find.

The code showed above will only give result where x=1. I have tried with different tolerance and alle three method: L-BFGS-B, TNC and SLSQP. This is the thread I have been looking at so far: Scipy.optimize: how to restrict argument values

How can I fix this?

I am using Spyder(Python 3.6).

Answer 1

You just encounterd the problem with local optimization: it strongly depends on the start (initial) values you pass in. If you supply [2, 1] it will find the correct minima.

Common solutions are:

• use your optimization in a loop with random starting points inside your boundaries

  import numpy as np
  from numpy import *; from scipy.optimize import *; from math import *
  
  def f(X):
      x=X[0];    y=X[1]
      return x**4-3.5*x**3-2*x**2+12*x+y**2-2*y
  
  bnds = ((1,3), (0, 2))
  
  for i in range(100):
  
      x_init = np.random.uniform(low=bnds[0][0], high=bnds[0][1])
      y_init = np.random.uniform(low=bnds[1][0], high=bnds[1][1])
  
      min_test = minimize(f,[x_init, y_init], bounds = bnds)
  
      print(min_test.x, min_test.fun)
  

• use an algorithm that can break free of local minima, I can recommend scipy's basinhopping()

• use a global optimization algorithm and use it's result as initial value for a local algorithm. Recommendations are NLopt's DIRECT or the MADS algorithms (e.g. NOMAD). There is also another one in scipy, shgo, that I have no tried yet.