$\newcommand{\bm}[1]{\boldsymbol{#1}} \newcommand{\textm}[1]{\textsf{#1}} \newcommand{\textnormal}[1]{\textsf{#1}} \def\T{{\mkern-2mu\raise-1mu\mathsf{T}}} \newcommand{\R}{\mathbb{R}} % real numbers \newcommand{\E}{{\rm I\kern-.2em E}} \newcommand{\w}{\bm{w}} % bold w \newcommand{\bmu}{\bm{\mu}} % bold mu \newcommand{\bSigma}{\bm{\Sigma}} % bold mu \newcommand{\bigO}{O} %\mathcal{O} \renewcommand{\d}[1]{\operatorname{d}\!{#1}}$

Exercises

Exercise 11.1 (Change of variable) Show why $\bSigma\bm{x} = \bm{b}/\bm{x}$ can be equivalently solved as $\bm{C}\bm{x} = \bm{b}/\bm{x}$, where $\bm{C}$ is the correlation matrix defined as $\bm{C} = \bm{D}^{-1/2}\bSigma\bm{D}^{-1/2}$ with $\bm{D}$ a diagonal matrix containing $\textm{diag}(\bSigma)$ along the main diagonal. Would it be possible to use instead $\bm{C} = \bm{M}^{-1/2}\bSigma\bm{M}^{-1/2}$, where $\bm{M}$ is not necessarily a diagonal matrix?

Exercise 11.2 (Naive diagonal risk parity portfolio) If the covariance matrix is diagonal, $\bSigma = \bm{D}$, then the system of nonlinear equations $\bSigma\bm{x} = \bm{b}/\bm{x}$ has the closed-form solution $\bm{x} = \sqrt{\bm{b}/\textm{diag}(\bm{D})}$. Explore whether a closed-form solution can be obtained for the rank-one plus diagonal case $\bSigma = \bm{u}\bm{u}^\T + \bm{D}$.

Exercise 11.3 (Vanilla convex risk parity portfolio) The solution to the formulation $$\begin{array}{ll} \underset{\bm{x}\ge\bm{0}}{\textm{maximize}} & \bm{b}^\T\log(\bm{x})\\ \textm{subject to} & \sqrt{\bm{x}^\T\bSigma\bm{x}} \le \sigma_0 \end{array}$$ is $$\lambda\bSigma\bm{x} = \bm{b}/\bm{x} \times \sqrt{\bm{x}^\T\bSigma\bm{x}}.$$ Can you solve for $\lambda$ and rewrite the solution in a more compact way without $\lambda$?

Exercise 11.4 (Newton's method) Newton’s method requires computing the direction $\bm{d} = \mathsf{H}^{-1}\nabla f$ or, equivalently, solving the system of linear equations $\mathsf{H}\,\bm{d} = \nabla f$ for $\bm{d}$. Explore whether a more efficient solution is possible by exploiting the structure of the gradient and Hessian: $$\begin{aligned} \nabla f &= \bSigma\bm{x} - \bm{b}/\bm{x},\\ \mathsf{H} &= \bSigma + \textm{Diag}(\bm{b}/\bm{x}^2). \end{aligned}$$

Exercise 11.5 (MM algorithm) The MM algorithm requires the computation of the largest eigenvalue $\lambda_\textm{max}$ of matrix $\bSigma$, which can be obtained from the eigenvalue decomposition of the matrix. A more efficient alternative is the power iteration method. Program both methods and compare their computational complexity.

Exercise 11.6 (Coordinate descent vs. SCA methods) Consider the vanilla convex formulation $$\begin{array}{ll} \underset{\bm{x}\ge\bm{0}}{\textm{minimize}} & \frac{1}{2}\bm{x}^\T\bSigma\bm{x} - \bm{b}^\T\log(\bm{x}). \end{array}$$ Implement the cyclical coordinate descent method and the parallel SCA method in a high-level programming language (e.g., R, Python, Julia, or MATLAB) and compare the convergence against the CPU time for these two methods. Then, re-implement these two methods in a low-level programming language (e.g., C, C++, C#, or Rust) and compare the convergence again. Comment on the difference observed.