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.

翻译：压缩感知（CS）涵盖了一系列用于在已知部分系数条件下恢复信号的理论与应用技术，其应用领域包括数学、物理、工程学以及多种医学科学。受我们对磁共振成像（MRI）与压缩感知数学机理研究兴趣的驱动，本文采用凸分析技术，解析性确定带有不等式约束优化问题（具体为具有体素权重的加权LASSO）中等效拉格朗日乘子的表达式。我们在关于保真项$\Vert{Ax-b}\Vert_2^2$的假设条件下研究该问题，这些假设涉及该梯度符号或其矩阵的正交性条件。更精确地说，我们要求$2(Ax-b)^TA$每个坐标的符号在原点的一个矩形邻域内固定（该矩形的边长取决于约束条件），或假设$A^TA$为对角矩阵。本文旨在探索拉格朗日乘子与加权LASSO变体约束条件之间的关系，特别关注可显式计算该关系的上述情形。由于拉格朗日乘子通过缩放加权LASSO的正则化项充当其调优参数，自然引发一个问题：它们是否可在MR图像重建与去噪等应用中作为有效的调优参数。本研究即为该方向的初步探索。