Scipy Minimize Constrained Function

I am solving the following optimization problem: with this Python code: from scipy.optimize import minimize import math def f(x): return math.log(x[0]**2 + 1) + x[1]**4 + x[0

Solution 1:

The issue seems to be with your initial guess. If I change your starting values to

x0 = [1.0, 1.0, 1.0]

Then your code will execute fine (at least on my machine)

Python 3.5.1 (v3.5.1:37a07cee5969, Dec 6 2015, 01:54:25) [MSC v.1900 64 bit (AMD64)] on win32

message:'Optimization terminated successfully.'njev:10jac:array([1.,0.,1.,0.])fun:0.6931471805582502nit:10status:0x:array([1.00000000e+00,-1.39724765e-06,1.07686548e-14])success:Truenfev:51

Solution 2:

Scipy's optimize module has lots of options. See the documentation or this tutorial. Since you didn't specify the method here, it will use Sequential Least SQuares Programming (SLSQP). Alternatively, you could use the Trust-Region Constrained Algorithm (trust-const).

For this problem, I found that trust-const seemed much more robust to starting values than SLSQP, handling starting values from [-2,-2,-2] to [10,10,10], although negative initial values resulted in increased iterations, as you'd expect. Negative values below -2 exceeded the max iterations, although I suspect might still converge if you increased max iterations, although specifying negative values at all for x1 and x3 is kind of silly, of course, I just did it to get a sense of how robust it was to a range of starting values.

The specifications for SLSQP and trust-const are conceptually the same, but the syntax is a little different (in particular, note the use of NonlinearConstraint).

from scipy.optimize import minimize, NonlinearConstraint, SR1

def f(x):
    return math.log(x[0]**2 + 1) + x[1]**4 + x[0]*x[2]

constr_func = lambda x: np.array( [ x[0]**3 - x[1]**2 - 1,
                                    x[0],
                                    x[2] ] )

x0=[0.,0.,0.]

nonlin_con = NonlinearConstraint( constr_func, 0., np.inf )

res = minimize( f, x0, method='trust-constr',
                jac='2-point', hess=SR1(),
                constraints = nonlin_con )

Here are the results, edited for conciseness:

fun: 0.6931502233468916
message: '`gtol` termination condition is satisfied.'
      x: array([1.00000063e+00, 8.21427026e-09, 2.40956900e-06])

Note that the function value and x values are the same as in @CoryKramer's answer. The x array may look superficially different at first glance, but both answers round to [1, 0, 0].