# Markowitz portfolio optimization - A multi-objective quadratic programming example
#
# In Markowitz portfolio optimization there are two objectives: to maximize reward
# while minimizing risk (i.e. variance). This example plots several points on the
# optimal frontier using a blended multi-objective approach, and shows that a point
# computed using a lexicographic approach also lies on this frontier.
#
# (C) 2024-2025 Fair Isaac Corporation
import xpress as xp
import numpy as np
from matplotlib import pyplot as plt
# The historical mean return on investment for five stocks
returns = np.array([0.31, 0.87, 0.31, 0.66, 0.24])
# The historical covariances of the five stocks
covariance = np.array([
[0.32, 0.70, 0.19, 0.52, 0.16],
[0.70, 4.35, -0.48, -0.06, -0.03],
[0.19, -0.48, 0.98, 1.10, 0.10],
[0.52, -0.6, 1.10, 2.48, 0.37],
[0.16, -0.3, 0.10, 0.37, 0.31]
])
p = xp.problem()
# Non-negative variables represent percentage of capital to invest in each stock.
x = p.addVariables(len(returns))
# All objectives must be linear - define a free transfer variable for the variance.
variance = p.addVariable(lb=-xp.infinity)
ctrs = [
xp.Sum(x) == 1, # Must invest 100% of capital.
xp.Dot(x, covariance, x) - variance <= 0 # Set up transfer var. for variance.
]
p.addConstraint(ctrs)
p.addObjective(xp.Dot(x, returns)) # Maximize mean return.
p.addObjective(variance) # Minimize variance.
p.setOutputEnabled(False)
# Vary the objective weights to explore the optimal frontier.
weights = np.linspace(0.05, 0.95, 100)
means = []
variances = []
for w in weights:
# First objective defines the sense of the problem.
p.setObjective(objidx=0, weight=w,sense=xp.maximize)
# Reverse the sense by assigning a negative weight because we minimize variance.
p.setObjective(objidx=1, weight=w-1)
p.optimize()
means.append(xp.Dot(p.getSolution(x), returns).item())
variances.append(p.getSolution(variance))
# Now we will maximize profit alone, and then minimize variance while not
# sacrificing more than 10% of the maximum profit.
p.setObjective(objidx=0, priority=1, weight=1, reltol=0.1,sense=xp.maximize)
p.setObjective(objidx=1, priority=0, weight=-1)
p.optimize()
m0 = xp.Dot(p.getSolution(x), returns).item()
v0 = p.getSolution(variance)
# Plot the optimal frontier and mark the final point that we calculated.
plt.plot(means, variances)
plt.plot(m0, v0, c='r', marker='.')
plt.title('Return on investment vs variance')
plt.xlabel('Expected return')
plt.ylabel('Variance')
plt.show()
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