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I need to invert a non-square matrix A under the constraint that the absolute value of each component of the solution is less than some maximum. In other words, I want [tex]\vec{b}[/tex] such that [tex]A . \vec b = \vec c[/tex] and [tex]|b_i| < \alpha[/tex].

Are there any established methods for doing this?

My idea is to start with the pseudoinverse to compute the unconstrained solution, and to then add components [tex]\vec v_i[/tex] from the null space of A to satisfy the constraint. If this doesn't help, I wanted to consecutively add more basis vectors from the singular value decomposition of A, starting with the basis corresponding to the smallest singular value.

However, I don't see how to pick the coefficients of the [tex]\vec v_i[/tex] in such a way that the final solution is optimal in the sense that there is no other solution that satiesfies the constraints but has a smaller [tex]||A.\vec b - \vec c||[/tex].

Anyone able to help? Pointers to appropriate literature would be appreciated as well, I am not quite sure what keywords I should be looking for.