Local Error Bounds for Affine Variational Inequalities on Hilbert Spaces

This paper gives some results related to the research problem about infinite-dimensional affine variational inequalities raised by N.D. Yen and X. Yang [Affine variational inequalities on normed spaces, J. Optim. Theory Appl., 178 (2018), 36--55]. Namely, we obtain local error bounds for affine variational inequalities on Hilbert spaces. To do so, we revisit two fundamental properties of polyhedral mappings. Then, we prove a locally upper Lipschitzian property of the inverse of the residual mapping of the infinite-dimensional affine variational inequality under consideration. Finally, we derive the desired local error bounds from that locally upper Lipschitzian property

Properties of Generalized Polyhedral Convex Set-Valued Mappings

This paper presents a study of generalized polyhedral convexity under basic operations on set-valued mappings. We address the preservation of generalized polyhedral convexity under sums and compositions of set-valued mappings, the domains and ranges of generalized polyhedral convex set-valued mappings, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results obtained provide a framework for representing the relative interior of the graph of a generalized polyhedral convex set-valued mapping in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex set-valued mappings from Banach spaces to locally convex topological vector spaces.Comment: 23 page

Relative Well-Posedness of Constrained Systems with Applications to Variational Inequalities

The paper concerns foundations of sensitivity and stability analysis, being primarily addressed constrained systems. We consider general models, which are described by multifunctions between Banach spaces and concentrate on characterizing their well-posedness properties that revolve around Lipschitz stability and metric regularity relative to sets. The enhanced relative well-posedness concepts allow us, in contrast to their standard counterparts, encompassing various classes of constrained systems. Invoking tools of variational analysis and generalized differentiation, we introduce new robust notions of relative coderivatives. The novel machinery of variational analysis leads us to establishing complete characterizations of the relative well-posedness properties with further applications to stability of affine variational inequalities. Most of the obtained results valid in general infinite-dimensional settings are also new in finite dimensions.Comment: 25 page

Affine variational inequalities on normed spaces

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