# CVXPY library

## CVXPY library

1. Constrained linear least squares Solve the following problem with cvxpy library.

\begin{split} &\|X \theta - y\|^2_2 \to \min\limits_{\theta \in \mathbb{R}^{n} } \\ \text{s.t. } & 0_n \leq \theta \leq 1_n \end{split}

2. Linear programming A linear program is an optimization problem with a linear objective and affine inequality constraints. A common standard form is the following:

\begin{array}{ll} \text{minimize} & c^Tx \\ \text{subject to} & Ax \leq b. \end{array}

Here A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m, and c \in \mathbb{R}^n are problem data and x \in \mathbb{R}^{n} is the optimization variable. The inequality constraint Ax \leq b is elementwise. Solve this problem with cvxpy library.

3. List the installed solvers in cvxpy using cp.installed_solvers() method.

4. Solve the following optimization problem using CVXPY:

\begin{array}{ll} \text{minimize} & |x| - 2\sqrt{y}\\ \text{subject to} & 2 \geq e^x \\ & x + y = 5, \end{array}

where x,y \in \mathbb{R} are variables. Find the optimal values of x and y.

5. Risk budget allocation Suppose an amount x_i>0 is invested in n assets, labeled i=1,..., n, with asset return covariance matrix \Sigma \in \mathcal{S}_{++}^n. We define the risk of the investments as the standard deviation of the total return

R(x) = (x^T\Sigma x)^{1/2}.

We define the (relative) risk contribution of asset i (in the portfolio x) as

\rho_i = \frac{\partial \log R(x)}{\partial \log x_i} = \frac{\partial R(x)}{R(x)} \frac{x_i}{\partial x_i}, \quad i=1, \ldots, n.

Thus \rho_i gives the fractional increase in risk per fractional increase in investment i. We can express the risk contributions as

\rho_i = \frac{x_i (\Sigma x)_i} {x^T\Sigma x}, \quad i=1, \ldots, n,

from which we see that \sum_{i=1}^n \rho_i = 1. For general x, we can have \rho_i <0, which means that a small increase in investment i decreases the risk. Desirable investment choices have \rho_i>0, in which case we can interpret \rho_i as the fraction of the total risk contributed by the investment in asset i. Note that the risk contributions are homogeneous, i.e., scaling x by a positive constant does not affect \rho_i.

• Problem statement: In the risk budget allocation problem, we are given \Sigma and a set of desired risk contributions \rho_i^\mathrm{des}>0 with \bf{1}^T \rho^\mathrm{des}=1; the goal is to find an investment mix x\succ 0, \bf{1}^Tx =1, with these risk contributions. When \rho^\mathrm{des} = (1/n)\bf{1}, the problem is to find an investment mix that achieves so-called risk parity.

• a) Explain how to solve the risk budget allocation problem using convex optimization. Hint. Minimize (1/2)x^T\Sigma x - \sum_{i=1}^n \rho_i^\mathrm{des} \log x_i.

• b) Find the investment mix that achieves risk parity for the return covariance matrix \Sigma below.

import numpy as np
import cvxpy as cp
Sigma = np.array(np.matrix("""6.1  2.9  -0.8  0.1;
2.9  4.3  -0.3  0.9;
-0.8 -0.3   1.2 -0.7;
0.1  0.9  -0.7  2.3"""))
rho = np.ones(4)/4