MATLAB Code for Solving L2-L1 and L2-TV Optimization Problems Using Interior Point Methods

In this document, we explore MATLAB code implementation for solving L2-L1 and L2-TV optimization problems using interior point methods. First, we need to understand the definitions and applications of these optimization problems. Then, we introduce the interior point method-based solution approach and explain its implementation in MATLAB code. Finally, we provide practical examples to help readers better understand how to use this code for problem-solving.

L2-L1 and L2-TV optimization problems are widely used in signal and image processing applications. The L2-L1 problem is a linear regression problem that aims to minimize the L1-norm of residuals while keeping the solution's L2-norm small. The L2-TV problem is an image restoration problem designed to minimize the total variation of an image while maintaining a small L2-norm for the solution.

The interior point method-based solution provides an efficient approach for solving these problems. This method utilizes the Iteratively Reweighted Least Squares (IRLS) algorithm, where each iteration operates on interior point sets and updates only these points. Key advantages include fast convergence rates and capability to handle large-scale problems. Implementing interior point methods in MATLAB requires mathematical and programming expertise, but once mastered, it enables efficient resolution of L2-L1 and L2-TV optimization challenges.

In the following example, we demonstrate image restoration using interior point methods. The damaged image will be recovered through L2-L1 and L2-TV optimization formulations. We explain the MATLAB implementation process with complete code reference, including key functions for weight matrix updates, convergence criteria checking, and proximal operator implementations for total variation regularization.