P. E. Farrell, M. Croci, T. M. Surowiec, Deflation for semismooth equations, Optimization Methods and Software (2019). Article.

Variational inequalities can in general support distinct solutions. In this paper we study an algorithm for computing distinct solutions of a variational inequality, with- out varying the initial guess supplied to the solver. The central idea is the combi- nation of a semismooth Newton method with a deflation operator that eliminates known solutions from consideration. Given one root of a semismooth residual, defla- tion constructs a new problem for which a semismooth Newton method will not con- verge 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 New- ton method. We demonstrate its utility on various finite- and infinite-dimensional examples drawn from constrained optimization, game theory, economics and solid mechanics.