Minimize function with SciPy

[PierreLCV]

Hello everyone,
I'm currently trying to minimize a function using a Python algorithm. The function is a bit complicated and requires an FEA solver to be calculated. Anyway, I was able to define my function in the classical way according to the parameter I'm trying to optimize:

def fonction_calcul(Parametre1):
      blabla function definition
      return function value

As this last function definition works well, I can easily get the values of my calculation_function for the parameter values I want. Knowing this, I thought I could then "foolishly" use scipy.optimize to find the minimization value of my function, but alas, it doesn't work!
Here's what I get starting with x=1

x_start=1.0
>>>	
result = spo.minimize(fonction_calcul, x_start)
>>>	
result
      fun: 106.69134802297765
 hess_inv: array([[1]])
      jac: array([0.])
  message: 'Optimization terminated successfully.'
     nfev: 2
      nit: 0
     njev: 1
   status: 0
  success: True
        x: array([1.])

Overall, as you can see, minimize calculates a single value and stops at this value, considering that it has optimized well (knowing that in this case, the minimization value is around 0.03...).

I've tried minimize with a "classic" function (polynomial) and it works well for finding the minimization value...

Any idea how to make minimize function in my case?

Thanks a lot!
Pierre

[paul18fr]

I guess you're using python to:

• retrieve results from your FEA solver
  
• to calculate the cost function value
  
• to start the optmization process (with constraints?)
  

Nontheless here I would suggest you to have a look to a powerfull optimizer which is Dakota inculing a lot of solvers (determinist & stochastic methods for instance); I wonder if now a python api exists (to be confirmed - i used it a long time ago).

Paul

[PierreLCV]

Dear paul,
Thank you for your reply ! I will take a look at Dakota :)
However, i really would like to understand why minimize (which appears to be a basic function) is not working...

Regards,
Pierre

[paul18fr]

here is the well known Rosembrock test case: 2 variables, unconstraint problem, single (global) minimum.

Keep in mind that:

• several evaluations/iterations are necessary; in your case, 1 iteration = 1 FEA
  
• gradient/hessian are numerically estimated (1 estimation = 1 simulation)
  
• your cost function is essential to succeed
  
• of course you can add constraints to bracket variables
  

Hope it helps.

from scipy.optimize import minimize
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html


def Rosembrock(x):   
    # f(x, y) = (1 - x)**2 + 100*(y - x**2)**2
    # minimum in (x, y) = (1, 1)
    return (1 - x[0])**2 + 100*(x[1] - x[0]**2)**2


# Method = 'NM'     # "Nelder-Mead"
# Method = 'BFGS'   # "Broyden-Fletcher-Goldfarb-Shanno"
# Method = 'CG'     # "Conjugate-gradient"
Method = 'SLSQP'    # "Sequential Least SQuares Programming"

# Starting point
x0 = [2., 3]

if (Method == 'NM'):
    # order 0 method (Nelder-Mead)
    #x0 = starting point
    
    Results = minimize(Rosembrock, 
                       x0, 
                       method='nelder-mead',
                       options={'xatol': 1e-8, 'disp': False}
                       )

    
else:
    # order 1 gradient based method (BFGS / CG / SLSQP)
    # Gradient and Hessian are numerically calculated => several evaluations are needed
    Results = minimize(Rosembrock, 
                       x0, 
                       method = Method,
                       jac = '3-point',
                       options={'disp': False}
                       )

# Optimization results:
print(f"Optimized value: {Results.x}")
print(f"Cost function value after optimization: {Results.fun}")
print(f"Number of iterations: {Results.nit}")
print(f"Number of function evalutaions: {Results.nfev}")
print(f"Status: {Results.message}")