Intro

OpenSoT provides an API to setup and solve generic Least-Squares like optimization problems in the form:

$$\begin{array}{rl}& \underset{\mathbf{x}}{min}\Vert \mathbf{A}\mathbf{x}-\mathbf{b}{\Vert }_{\mathbf{W}}+{\mathbf{c}}^T\mathbf{x}+ϵ\Vert \mathbf{x}\Vert \\ & s.t.{\mathbf{d}}_m\le \mathbf{C}\mathbf{x}\le {\mathbf{d}}_M\end{array}$$

with $\mathbf{x}\in {\mathbb{R}}^n$, $\mathbf{A}\in {\mathbb{R}}^{m\times n}$, $\mathbf{b}\in {\mathbb{R}}^m$, $\mathbf{W}\in {\mathbb{R}}^{m\times m}$, $\mathbf{c}\in {\mathbb{R}}^n$, $\mathbf{C}\in {\mathbb{R}}^{l\times n}$, ${\mathbf{d}}_m$ and ${\mathbf{d}}_M\in {\mathbb{R}}^l$, and $ϵ\in {\mathbb{R}}^{+}$. This optimization problem is equivalent to the following QP problem:

$$\begin{array}{rl}& \underset{\mathbf{x}}{min}{\mathbf{x}}^T\mathbf{H}\mathbf{x}+{\mathbf{g}}^T\mathbf{x}\\ & s.t.{\mathbf{d}}_m\le \mathbf{C}\mathbf{x}\le {\mathbf{d}}_M\end{array}$$

with $\mathbf{H}={\mathbf{A}}^T\mathbf{W}\mathbf{A}+ϵ\mathbf{I}$ and ${\mathbf{g}}^T={\mathbf{c}}^T-2{\mathbf{b}}^T\mathbf{W}\mathbf{A}$, and can be solved by dedicated solvers.