Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems. (3rd August 2017)

• Record Type: Journal Article
• Authors: Gwinner, Joachim
• Is Part Of: Optimization. Volume 66:Number 8(2017)
• Page Start: 1323
• Page End: 1336
• Publication Date: 2017-08-03
• Subjects: Primal variational inequality -- mixed formulation -- Brezzi splitting theorem -- Babuška–Brezzi condition -- Lagrange multipliers -- Coulomb/Tresca frictional unilateral contact problem; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2016.1262863
• Languages: English
• ISSNs: 0233-1934

Abstract

Abstract: This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.