# Deflation for semismooth equations

arXiv: https://arxiv.org/abs/1904.13299 · HTML: https://mlsanity.com/paper/1904.13299/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2019.

## Built on by (most cited first, 2 of 2)

- [Simplifying deflation for non-convex optimization with applications in Bayesian inference and topology optimization](https://mlsanity.com/paper/2201.11926/) (2022)
- [Creating walls to avoid unwanted points in root finding and optimization](https://mlsanity.com/paper/2309.11475/) (2023)

## Abstract (arXiv)

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, without varying the initial guess supplied to the solver. The central idea is the combination of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, deflation constructs a new problem for which a semismooth Newton method will not converge to the known root, even from the same initial guess. This enables the discovery of other roots. We prove the effectiveness of the deflation technique under the same assumptions that guarantee locally superlinear convergence of a semismooth Newton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.