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pythonoptimizationcvxpy

cvxpy.error.DCPError: Problem does not follow DCP rules

Getting my head around the DCP rules. I am looking at the portfolio optimisation example provided on the CVXPY website (see below original codes). Had a look at some of the other queries that deals with DCP rules but couldn't get the answer I wanted.

I tried replacing the Sigma (i.e. covariance) in their code (which is randomly generated) with cov generated from some historic returns for some of the asset classes. Everything else is the same. Yet I get cvxpy.error.DCPError: Problem does not follow DCP rules.

I have also added pics of the two Sigmas (one generated randomly by the CVXPY code and the other Sigma(1) is the historic cov array I use)

Both are 9*9 arrays, but as I mentioned replacing the randomly generated array with an array with historic numbers gives me that error, all other codes kept the same. Any idea what's causing this issue?


# Generate data for long only portfolio optimization.
import numpy as np
import pandas as pd
import matplotlib
matplotlib.use('TkAgg')
import matplotlib.pyplot as plt

np.random.seed(1)
n = 10
mu = np.abs(np.random.randn(n, 1))
Sigma = np.random.randn(n, n)
Sigma = Sigma.T.dot(Sigma)


# Long only portfolio optimization.
import cvxpy as cp


w = cp.Variable(n)
gamma = cp.Parameter(nonneg=True)
ret = mu.T*w
risk = cp.quad_form(w, Sigma)
prob = cp.Problem(cp.Maximize(ret - gamma*risk),
               [cp.sum(w) == 1,
                w >= 0])


# Compute trade-off curve.
SAMPLES = 100
risk_data = np.zeros(SAMPLES)
ret_data = np.zeros(SAMPLES)
gamma_vals = np.logspace(-2, 3, num=SAMPLES)
for i in range(SAMPLES):
    gamma.value = gamma_vals[i]
    prob.solve()
    risk_data[i] = cp.sqrt(risk).value
    ret_data[i] = ret.value


# Plot long only trade-off curve.
import matplotlib.pyplot as plt
#%matplotlib inline
#%config InlineBackend.figure_format = 'svg'

markers_on = [29, 40]
fig = plt.figure()
ax = fig.add_subplot(111)
plt.plot(risk_data, ret_data, 'g-')
for marker in markers_on:
    plt.plot(risk_data[marker], ret_data[marker], 'bs')
    ax.annotate(r"$\gamma = %.2f$" % gamma_vals[marker], xy=(risk_data[marker]+.08, ret_data[marker]-.03))
for i in range(n):
    plt.plot(cp.sqrt(Sigma[i,i]).value, mu[i], 'ro')
plt.xlabel('Standard deviation')
plt.ylabel('Return')
plt.show()

enter image description here enter image description here

Solution

  • Observation: it is possible that the variance-covariance matrix in a portfolio optimization is not positive semi-definite. In theory the convariance matrix can be shown to be positive semi-definite. However, due to floating-point rounding errors, we may actually see (slightly) negative eigenvalues. (Note: a positive semi-definite matrix has non-negative eigenvalues).

    I know of three approaches to handle this:

    1. Use a statistical technique called shrinkage,
    2. Perturb the diagonal a bit by adding small constant to each diagonal element,
    3. Use a variant of the standard portfolio model based on mean adjusted returns.

    For details see link.