Analogy QP and Pseudoinverse

Question

It is well known that the pseudo inverse of an $m \times n$ matrix $A$ automatically minimizes $\frac{1}{2}x^Tx$ if $m<n$ and $\frac{1}{2}(b-Ax)^T(b-Ax)$ if $m<n$ for the linear system $Ax=b$. My question is: If $x^m\leq x \leq x^M$, can we formulate the solution for $x$ with the same optimality conditions as of the pseudoinverse in ONE QP problem?