CVXPY: How to maximize dot product of two vectors

Question

Suppose we have three features and 252 samples per each feature. Here, features are returns of three different stocks. The goal is to maximize the total return, i,e,

My question is how to define this objective function inside a CVXpy optimization problem?

The procedural view of the program should be look like this:

import numpy as np
import cvxpy as cvx;

return1 = np.random.normal(loc=0.05, scale=0.25, size=252)
return2 = np.random.normal(loc=0.01, scale=0.2, size=252)
return3 = np.random.normal(loc= -0.05, scale=0.15, size=252)
returns = np.array([return1, return2, return3])

# number of stocks m is number of rows of returns
m = returns.shape[0]

# x variables (to be found with optimization)
x = cvx.Variable(m)

# portfolio return
portfolio_return = cvx.multiply(returns, x)

#objective function
objective = cvx.Minimize(-portfolio_return)

#constraints
constraints = [x >= 0, sum(x) == 1]

#use cvxpy to solve the objective
cvx.Problem(objective, constraints).solve()

#retrieve the weights of the optimized portfolio
x_values = x.value

print(x_values)

However, it returns an error:

ValueError: Cannot broadcast dimensions  (3, 252) (3,)

When we represent x as x = cvx.Variable(shape=(m,1)) we get another error

ValueError: The 'minimize' objective must resolve to a scalar.

Python Version: 3.8.8

CVXPY Version: 1.1.12

Your `returns` is a matrix. It should be a vector (the same length as x). — Erwin Kalvelagen, May 16 '21 at 14:14
@sci9, just found this https://www.javacodexamples.com/examples/cvxpy-how-to-maximize-dot-product-of-two-vectors. Was it posted by you? Or is this a case of blatant plagiarism? — William R. Ebenezer, Feb 13 '22 at 20:26

sci9 · Accepted Answer · 2022-03-13T14:04:33.660

Thanks to cvxpy community, I've reformulated the problem and it now runs without any error.

import numpy as np
import cvxpy as cvx;
return1 = np.random.normal(loc=0.05, scale=0.25, size=252)
return2 = np.random.normal(loc=0.01, scale=0.2, size=252)
return3 = np.random.normal(loc= -0.05, scale=0.15, size=252)
returns = np.array([return1, return2, return3])
m = returns.shape[0]
cov = np.cov(returns)
x = cvx.Variable(m)
portfolio_return = returns.mean(axis=1)
objective = cvx.Maximize(x @ portfolio_return)
constraints = [x >= 0, sum(x) == 1]
cvx.Problem(objective, constraints).solve()
x_values = x.value
x_values.round(2)

CVXPY: How to maximize dot product of two vectors

1 Answers1