Find a vector that is as proportional as possible to a given vector under a set of linear constraints

Question

Let $d\in \mathbb{R}^n, \ b\in \mathbb{R}^n, \ A \in \mathbb{R}^{m\times n},\ \lambda\in \mathbb{R}$.

Let $x=\lambda d+\varepsilon $, where $\varepsilon\in \mathbb{R}^n$.

Let $E_\lambda =\left \{\varepsilon\in \mathbb{R}\ \vert \ Ax\leq b \right\}$.

Find $\varepsilon_o\in E_\lambda$ such that $(\forall \varepsilon^{\prime} \in E_\lambda) \Vert \varepsilon_o \Vert \le \Vert\varepsilon^{\prime}\Vert$.

Answer 1

So $d, b, A, \lambda$ are fixed and $x$ is variable. You can solve this as a quadratic programming problem where you minimize $(x-\lambda d)^2$ subject to $Ax \le b$.