Deflation for semismooth equations
For researchers solving variational inequalities with multiple solutions, this method enables discovery of alternative solutions without manual intervention, though it is an incremental extension of existing deflation and Newton methods.
The paper introduces a deflation technique combined with a semismooth Newton method to compute multiple distinct solutions of variational inequalities from the same initial guess, proving its effectiveness under assumptions for superlinear convergence and demonstrating it on examples from optimization, game theory, economics, and solid mechanics.
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.