# A Variational Inequality Model for the Construction of Signals from Inconsistent Nonlinear Equations

arXiv: https://arxiv.org/abs/2105.07380 · HTML: https://mlsanity.com/paper/2105.07380/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 May 2021.

## Built on by (most cited first, 1 of 1)

- [Learning truly monotone operators with applications to nonlinear inverse problems](https://mlsanity.com/paper/2404.00390/) (2024)

## Abstract (arXiv)

Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies a number of nonlinear equations involving firmly nonexpansive operators. We investigate this formalism in the case when, due to inaccurate modeling or perturbations, the nonlinear equations are inconsistent. A relaxed formulation of the original problem is proposed in the form of a variational inequality. The properties of the relaxed problem are investigated and a provenly convergent block-iterative algorithm, whereby only blocks of the underlying firmly nonexpansive operators are activated at a given iteration, is devised to solve it. Numerical experiments illustrate robust recoveries in several signal and image processing applications.