Deflation for semismooth equations. (1st November 2020)

• Record Type: Journal Article
• Authors: Farrell, Patrick E.; Croci, Matteo; Surowiec, Thomas M.
• Is Part Of: Optimization methods and software. Volume 35:Number 6(2020)
• Page Start: 1248
• Page End: 1271
• Publication Date: 2020-11-01
• Subjects: Deflation -- semismooth Newton -- variational inequalities -- complementarity problems; 65K15 Numerical methods for variational inequalities and related problems -- 65P30 Bifurcation problems -- 65H10 Systems of equations -- 35M86 Nonlinear unilateral problems and nonlinear variational inequalities of mixed type -- 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions); Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2019.1613655
• Languages: English
• ISSNs: 1055-6788

Abstract

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.