# Solving the L1 regularized least square problem via a box-constrained smooth minimization

## TL;DR

This paper introduces a smooth convex reformulation of the L1 regularized least squares problem, enabling the use of fast optimization algorithms and providing insights into duality properties, with experimental validation.

## Contribution

It proposes a novel smooth minimization approach for L1 regularized least squares, demonstrating its effectiveness and theoretical properties, including duality relations.

## Key findings

- The smooth reformulation facilitates faster optimization methods.
- The dual of dual equals primal property holds for the problem.
- Experimental results validate the proposed approach's efficiency.

## Abstract

In this paper, an equivalent smooth minimization for the L1 regularized least square problem is proposed. The proposed problem is a convex box-constrained smooth minimization which allows applying fast optimization methods to find its solution. Further, it is investigated that the property "the dual of dual is primal" holds for the L1 regularized least square problem. A solver for the smooth problem is proposed, and its affinity to the proximal gradient is shown. Finally, the experiments on L1 and total variation regularized problems are performed, and the corresponding results are reported.

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Source: https://tomesphere.com/paper/1704.03443