We introduce augmented Lagrangian methods for solving finite dimensional variational inequality problems whose feasible sets are defined by convex inequalities, generalizing the proximal augmented Lagrangian method for constrained optimization. At each iteration, primal variables are updated by solving an unconstrained variational inequality problem, and then dual variables are updated through ...

Variational inequalities are an important mathematical tool for modelling free boundary problems that arise in different application areas. Due to the intricate nonsmooth structure of the resulting models, their analysis and optimization is a difficult task that has drawn the attention of researchers for several decades. In this paper we focus on a class of variational inequalities, called of t...

Abstract We study a minimisation problem in L p and ∞ for certain cost functionals, where the class of admissible mappings is constrained by Navier–Stokes equations. Problems this type are motivated variational data assimilation atmospheric flows arising weather forecasting. Herein we establish existence PDE-constrained minimisers all , also that converge to as → ∞. further show solve an Euler–...

Smoothing methods have become part of the standard tool set for the study and solution of nondifferentiable and constrained optimization problems as well as a range of other variational and equilibrium problems. In this note we synthesize and extend recent results due to Beck and Teboulle on infimal convolution smoothing for convex functions with those of X. Chen on gradient consistency for non...

Evolution mixed maximal monotone variational inclusions with optimal control, in reflexive Banach spaces, are analized. Solvability analysis is performed on the basis of composition duality principles. Applications to nonlinear diffusion constrained problems, as well as to quasistatic elastoviscoplastic contact problems exemplify the theory.

Variational inequalities and even quasi-variational inequalities, as means of expressing constrained equilibrium, have utilized geometric properties of convex sets, but the theory of tangent cones and normal cones has yet to be fully exploited. Much progress has been made in that theory in recent years in understanding the variational geometry of nonconvex as well as convex sets and applying it...

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves γ in a differentiable manifold M that are everywhere tangent to a smooth distribution D on M ; such curves are called horizontal. We study the manifold structure of the set ΩP,Q(M,D) of horizontal curves that join two submanifolds P and Q of M . We consider an ...