Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

Abstract

This paper deals with quasi-variational inequality problems (QVIs) in a generic Banach space setting. We provide a theoretical framework for the analysis of such problems which is based on two key properties: the pseudomonotonicity (in the sense of Brezis) of the variational operator and a Mosco-type continuity of the feasible set mapping. We show that these assumptions can be used to establish the existence of solutions and their computability via suitable approximation techniques. In addition, we provide a practical and easily verifiable sufficient condition for the Mosco-type continuity property in terms of suitable constraint qualifications.Based on the theoretical framework, we construct an algorithm of augmented Lagrangian type which reduces the QVI to a sequence of standard variational inequalities. A full convergence analysis is provided which includes the existence of solutions of the subproblems as well as the attainment of feasibility and optimality. Applications and numerical results are included to demonstrate the practical viability of the method.

Brezis Pseudomonotonicity is Strictly Weaker than Ky–Fan Hemicontinuity

“…Note that we call F bounded if it maps bounded sets in X to bounded sets in X * . [11,Corollary 4.2]). Let A ⊆ X be a nonempty, convex, weakly compact set, and F : X → X * a bounded pseudomonotone operator.…”

“…• If F : X → X * is pseudomonotone, then the solution set of the VI (1) is always weakly sequentially closed. More generally, if {x k } is a sequence of suitable "approximate" solutions of the VI, then every weak limit point of {x k } belongs to its solution set, see [11].…”

“…The standard application of pseudomonotonicity was the construction of existence results for VIs, a topic which occurs in all these references and was also revisited in [12]. In addition, pseudomonotonicity has turned out to be quite useful when analyzing convergence of iterative algorithms for constrained minimization and variational or quasi-variational inequalities, see [11,16] for more details.…”

Existence of Generalized Augmented Lagrange Multipliers for Constrained Optimization Problems

“…For example, the Hestenes-Powell-Rockafellar augmented Lagrangian [1][2][3], the cubic augmented Lagrangian [4], Mangasarian's augmented Lagrangian [5,6], the exponential penalty function [7,8], the log-sigmoid Lagrangian [9], modified barrier functions [8,10], the p-th power augmented Lagrangian [11], and nonlinear augmented Lagrangian functions [12][13][14][15]. The other related discussion on augmented Lagrangians regarding special constrained optimization includes second-order cone programming [16,17], semidefinite programming [18][19][20], cone programming [21][22][23], semi-infinite programming [24,25], min-max programming [26], distributed optimization [27], mixed integer programming [28], stochastic mixed-integer programs [29], generalized Nash equilibrium problems [30], quasi-variational inequalities [31], composite convex programming [32], and sparse discrete problems [33]. The duality theory is closely related to the perturbation of primal problem.…”

Nonsmooth dynamics of generalized Nash games

“…To the best of our knowledge this approach is new for the generalized Nash equilibrium problem. Additionally, it allows us to study the GNEP in an infinite-dimensional setting, thus extending the traditional Euclidean space as in [11,34], and consider a generic constraint set. Last but not least, this work brings new incentives in the study of these nonsmooth dynamical systems and offers some insights into existence of solutions state-of-the-art and the need for generalizations.…”