How to re-parametrize for quadratic minimization?

Question

Given a real-rectangular matrix $S$ and inorder to solve this simple quadratic programming problem:

Minimize $w'S'Sw = \|S w\|^2$ over $w$ subject to
$e^Tw = 1$ and $w \geq 0$

using a solver I want a re-parametrization of the problem to the form:

$min(-d^T b + 1/2 b^T D b)$ with the constraints $A^T b \geq b_0$

so that I can use a general purpose optimization software for quadratic programming.

Question: So now, what would $d,b,D,A,b_0$ be?

Secondly, how is this re-parametrization done (is there a well known procedural, aspect to this or is it just algebra)? I ask because, I would want to use this general purpose solver for various quadratic minimization programs.

I can see that $b$ is $w$ and I am guessing that $A$ is an identity matrix and $b_0$ is a vector of zeros. What is D?

Answer 1

Obviously, $d=0$, $b=w$ and $D=S^{*}S$. As for the equality constraint, $$ e^{T}w=1\iff e^{T}w\geq1\hspace{1em}\text{and}\hspace{1em}-e^{T}w\geq-1. $$ Therefore, your $A$ is (in block-matrix form) $$ A=\left[\begin{array}{c} e^{T}\\ -e^{T}\\ I \end{array}\right] $$ and your $b_{0}$ is $$ b_{0}=\left[\begin{array}{c} 1\\ -1\\ 0\\ \vdots\\ 0 \end{array}\right]. $$