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

On the determination of Lagrange Multipliers for a weighted LASSO problem using geometric and convex analysis techniques