Matrix Constraints with Scipy.Optimize.Minimize

Question

I'm trying out python's scipy.optimize to minimize some function using the SLSQP algorithm. The optimization seems to work fine unconstrained and with one matrix constraint, but then I get an error when I add a second matrix constraint.

import nlopt
import numpy as np
from scipy.optimize import minimize

n = 3
n_sim = 10000
risk_aversion = 5

Y = np.random.normal(loc=0, scale=1, size=(n_sim, n))

mu_Y = np.mean(Y, axis=0).reshape(1, n)
sigma_Y = np.cov(Y, rowvar=0, bias=1)

x0 = np.ones((n, 1)) / n

Aeq = np.ones((1, n))
beq = 1
A =  np.array([[1, 1, 0], [-1, -1, 0]])
b = np.array([[0.25], [-0.5]])

def func(x, mu, sigma, risk_aversion):
    return -np.dot(mu, x) + 0.5 * risk_aversion * np.dot(np.dot(x.T, sigma), x)
def func_deriv(x, sigma, risk_aversion):
    return risk_aversion * np.dot(sigma, x)

c_ = {'type': 'eq', 'fun' : lambda x: np.dot(Aeq, x) - beq, 'jac' : lambda x: Aeq}
b_ = [(0,1) for i in range(n)]

This version with one constraint and bounds works fine

res = minimize(lambda x: func(x, mu_Y, sigma_Y, risk_aversion), x0, 
               jac=lambda x: func_deriv(x, sigma_Y, risk_aversion), 
               constraints=c_, bounds=b_, method='SLSQP', options={'disp': True})

However, the two constraint version gives me an error

d_ = (c_,
        {'type': 'ineq', 'fun' : lambda x: np.dot(A, x) - b, 'jac' : lambda x: A})

res2 = minimize(lambda x: func(x, mu_Y, sigma_Y, risk_aversion), x0, 
                jac=lambda x: func_deriv(x, sigma_Y, risk_aversion), 
                constraints=d_, bounds=b_, method='SLSQP', options={'disp': True})

John · Accepted Answer · 2015-01-13 20:21:05Z

2

If I use

A =  np.array([-1, -1, 0])
b = -0.5

instead of what I had originally, then the optimization works. This suggests that I can't do a matrix of constraints (like I would with fmincon in Matlab), unless someone can confirm otherwise. However, I can just keep adding constraints such that it will work. The constraints have to be in the form of tuples of dictionaries, so I put together

d = []
for i in range(0, b.size):
    d.append({'type': 'ineq', 'fun' : lambda x: np.dot(A[i, ], x) - b[i], 'jac' : lambda x: A[i, ]})
d_ = tuple(d)
c_tuple = c_,
d_ = d_ + c_tuple

which seems to be working properly.

answered Jan 13, 2015 at 20:21

John

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2 Comments

pv. Over a year ago

When appending stuff to the constraint dict in a for loop, be careful about this: stackoverflow.com/questions/233673/lexical-closures-in-python

John Over a year ago

@pv My goal was more to get it to work well enough so that I could compare the basic performance to nlopt. nlopt was about 3x faster, so I'm going to focus more on using that (which does not require this syntax).

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