# Deflation for semismooth equations

## TL;DR

This paper introduces a deflation technique combined with semismooth Newton methods to find multiple solutions of variational inequalities without changing initial guesses, proven effective in diverse applications.

## Contribution

It presents a novel deflation approach that enables the discovery of multiple solutions in variational inequalities using a single initial guess.

## Key findings

- Deflation effectively eliminates known solutions from consideration.
- The method guarantees local superlinear convergence.
- Demonstrated success on problems from various fields.

## Abstract

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.