  Compressed Sensing (CS) encompasses a broad array of theoretical and applied
techniques for recovering signals, given partial knowledge of their
coefficients. Its applications span various fields, including mathematics,
physics, engineering, and several medical sciences. Motivated by our interest
in the mathematics behind Magnetic Resonance Imaging (MRI) and CS, we employ
convex analysis techniques to analytically determine equivalents of Lagrange
multipliers for optimization problems with inequality constraints, specifically
a weighted LASSO with voxel-wise weighting. We investigate this problem under
assumptions on the fidelity term $\Vert{Ax-b}\Vert_2^2$, either concerning the
sign of its gradient or orthogonality-like conditions of its matrix. To be more
precise, we either require the sign of each coordinate of $2(Ax-b)^TA$ to be
fixed within a rectangular neighborhood of the origin, with the side lengths of
the rectangle dependent on the constraints, or we assume $A^TA$ to be diagonal.
The objective of this work is to explore the relationship between Lagrange
multipliers and the constraints of a weighted variant of LASSO, specifically in
the mentioned cases where this relationship can be computed explicitly. As they
scale the regularization terms of the weighted LASSO, Lagrange multipliers
serve as tuning parameters for the weighted LASSO, prompting the question of
their potential effective use as tuning parameters in applications like MR
image reconstruction and denoising. This work represents an initial step in
this direction.
