Ill-posed linear inverse problems with box constraints: A new convex optimization approach

TL;DR

This paper introduces a novel convex optimization method for solving ill-posed linear inverse problems with box constraints, ensuring solutions lie within the feasible set and analyzing their data dependence.

Contribution

It proposes a new convex optimization approach using the dual of a moment generating function for problems with box constraints, providing interior solutions and theoretical insights.

Findings

Solutions lie in the interior of the constraint set

The method offers comparison results with existing approaches

Analysis of solution dependence on data and Le Chatellier principle

Abstract

Consider the linear equation $\mathbf{A}\mathbf{x}=\mathbf{y}$, where $\mathbf{A}$ is a $k\times N$-matrix, $\mathbf{x}\in\mathcal{K}\subset \mathbb{R}^N$ and $\mathbf{y}\in\mathbb{R}^M$ a given vector. When $\mathcal{K}$ is a convex set and $M\not= N$ this is a typical ill-posed, linear inverse problem with convex constraints. Here we propose a new way to solve this problem when $\mathcal{K} = \prod_j[a_j,b_j]$. It consists of regarding $\mathbf{A}\mathbf{x}=\mathbf{y}$ as the constraint of a convex minimization problem, in which the objective (cost) function is the dual of a moment generating function. This leads to a nice minimization problem and some interesting comparison results. More importantly, the method provides a solution that lies in the interior of the constraint set $\mathcal{K}$. We also analyze the dependence of the solution on the data and relate it to the Le Chatellier principle.