python - Constrained Optimization with Scipy for a nonlinear fucntion -

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i trying maximize x^(0.5)y^(0.5) st. x+y=10 using scipy.

i can't figure out method use. appreciate if guide me on this.

here 2 possible ways:

the first version uses fmin_cobyla , therefore not require derivative of f.

from scipy.optimize import fmin_cobyla  f = lambda x : - (x[0]**0.5 * x[1]**(0.5))  # x + y = 10 <=> (x + y - 10 >= 0) & (-x -y + 10 >= 0) c1 = lambda x: x[0] + x[1] - 10 c2 = lambda x: 10 - x[0] - x[1]  fmin_cobyla(f, [0,0], cons=(c1, c2))

and get: array([ 4.9999245, 5.0000755])

the second version uses fmin_slsqp , exploits can calculate partial derivatives analytically:

from scipy.optimize import fmin_slsqp  f = lambda x : - (x[0]**0.5 * x[1]**(0.5))  def f_prime(x):     ddx1 = 0.5 * x[0]**-0.5 * x[1]**0.5     ddx2 = 0.5 * x[1]**-0.5 * x[0]**0.5     return [ddx1, ddx2]  f_eq = lambda x: x[0] + x[1] - 10  fmin_slsqp(f, [0.01,0.01], fprime=f_prime, f_eqcons=f_eq)

this output:

optimization terminated successfully.    (exit mode 0)         current function value: -5.0         iterations: 2         function evaluations: 2         gradient evaluations: 2

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