\chapter{Introduction}\label{ch: introduction}

(P) is the linear least squares problem
\[\min_{w}\ \left\lVert \hat{X}w-\hat{y} \right\rVert\]
where
\[
  \hat{X} = 
  \begin{bmatrix} 
    X^T \\ 
    \lambda I_m 
  \end{bmatrix}, 
  \ \ 
  \hat{y} = 
  \begin{bmatrix} 
    y \\ 
    0 
  \end{bmatrix},
\]
with $X$ the (tall thin) matrix from the ML-cup dataset by prof. Micheli, $\lambda > 0$ and $y$ is a random vector.
\begin{itemize}
\item[--] (A1) is an algorithm of the class of limited-memory quasi-Newton methods.
\item[--] (A2) is a cothin QR factorization with Householder reflectors, in the variant where one does not form the matrix $Q$, but stores the Householder vectors $u_k$ and uses them to perform (implicitly) products with $Q$ and $Q^T$.
\end{itemize}
No off-the-shelf solvers allowed. In particular you must implement yourself the thin QR factorization, and the computational cost of your implementation should be at most quadratic in $m$.

\subsection*{Outline}
This report is organized as follows:
\begin{description}
\item[\autoref{ch: problem definition},] in which the problem is reformulated under the mathematical aspect;
\item[\autoref{ch: algorithms},] where we will include the implemented algorithms, with the analysis of convergence and complexity;
\item[\autoref{ch: experiments},] to evaluate and compare (A1) with (A2) for this task and provide different tests in order to examine deeper the algorithms;
\item[\autoref{ch: conclusion},] in which conclusions are drawn, offering a critical analysis of the results obtained.
\end{description}