  Consider the linear equation $\mathbf{A}\mathbf{x}=\mathbf{y}$, where
$\mathbf{A}$ is a $k\times N$-matrix, $\mathbf{x}\in\mathcal{K}\subset
\mathbb{R}^N$ and $\mathbf{y}\in\mathbb{R}^M$ a given vector. When
$\mathcal{K}$ is a convex set and $M\not= N$ this is a typical ill-posed,
linear inverse problem with convex constraints. Here we propose a new way to
solve this problem when $\mathcal{K} = \prod_j[a_j,b_j]$. It consists of
regarding $\mathbf{A}\mathbf{x}=\mathbf{y}$ as the constraint of a convex
minimization problem, in which the objective (cost) function is the dual of a
moment generating function. This leads to a nice minimization problem and some
interesting comparison results. More importantly, the method provides a
solution that lies in the interior of the constraint set $\mathcal{K}$. We also
analyze the dependence of the solution on the data and relate it to the Le
Chatellier principle.
