Optimize with SciPy vector to scalar function with constraint

I would need to optimize a function f with respects to a vector x, that takes as input a constant matrix m and returns a scalar v >= 0.

MWE with random numbers:

import numpy as np
from scipy.optimize import minimize

np.random.seed(1)

m = np.array([[1,0,0.15],[2,0,0.15],[1.5,0.2,0.2],[3,0.5,0.1],[2.2,0.1,0.15]])
x0 = np.random.rand(5)*2

def f(x, m):
    pg = -np.concatenate((-arr[:, :2], x.reshape(-1, 1)), axis=1).sum(axis=1)
    return sum(arr[:, 2] * pg)

res = minimize(
    f, x0,
    method='nelder-mead', args=(m,),
    options={'xatol': 1e-8, 'maxiter': 1e+4, 'disp': True}
)

How do I set up the constraint for the output value? As far as I read in the doc I can only set constraints for the inputs. I read this post saying to use minimize_scalar, but it can only be used when the input is scalar as well.

Answer

Simply add the constraint f(x,m) >= 0:

import numpy as np
from scipy.optimize import minimize

np.random.seed(1)

m = np.array([[1,0,0.15],[2,0,0.15],[1.5,0.2,0.2],[3,0.5,0.1],[2.2,0.1,0.15]])
x0 = np.random.rand(5)*2

def f(x, m):
    pg = -np.concatenate((-arr[:, :2], x.reshape(-1, 1)), axis=1).sum(axis=1)
    return sum(arr[:, 2] * pg)

# add the constraint f(x, m) >= 0
con = [{'type': 'ineq', 'fun': lambda x: f(x, m)}]

res = minimize(
    f, x0,
    constraints=con,
    method='nelder-mead', args=(m,),
    options={'xatol': 1e-8, 'maxiter': 1e+4, 'disp': True}
)

Alternatively, you can enforce a positive objective function value by minimizing some vector norm of your objective, e.g. f(x,m)**2. You wouldn’t need a constraint then.

PS: The second argument of your function should probably be arr instead of m.

PPS: Since both your objective function and the constraint are continuously differentiable, a gradient-based algorithm will very likely perform much better than Nelder-Mead, even if the gradient is approximated by finite differences.