If $a_{m\times 1}$ and $Q_{m\times n}$ ($m<n $) are known, and we know every element of $b$ is between $[-1\ \ 1]$, how to determine $b$ to minimize $\|a+Qb\|_2$?


closed as off-topic by Suvrit, Stefan Kohl, user1688, Wolfgang, Alex Degtyarev Sep 18 '16 at 21:09

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  • $\begingroup$ Solve it as a quadratic program (using an iterative algorithm). $\endgroup$ – Suvrit Sep 18 '16 at 12:43

There is an algorithm designed specifically for the OP's problem, given in Bounded-Variable Least-Squares: an Algorithm and Applications [pdf]:

BVLS (bounded-variable least-squares) is modelled on NNLS and solves the problem \begin{equation} \min_{l \leq x \leq u} \|A x - b\|_2, \end{equation} where $l,x,u \in \mathbb{R}^n, b \in \mathbb{R}^m$ and $A$ is an $m$ by $n$ matrix.


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