Local strong maximal monotonicity and full stability for parametric variational systems

The paper introduces and characterizes new notions of Lipschitzian and H\"olderian full stability of solutions to general parametric variational systems described via partial subdifferential and normal cone mappings acting in Hilbert spaces. These notions, postulated certain quantitative properties of single-valued localizations of solution maps, are closely related to local strong maximal monotonicity of associated set-valued mappings. Based on advanced tools of variational analysis and generalized differentiation, we derive verifiable characterizations of the local strong maximal monotonicity and full stability notions under consideration via some positive-definiteness conditions involving second-order constructions of variational analysis. The general results obtained are specified for important classes of variational inequalities and variational conditions in both finite and infinite dimensions

Stability analysis for parameterized variational systems with implicit constraints

In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non-restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examplesComment: 26 page

Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems

The approach to the theory of many-particle interacting systems from a unified standpoint, based on the variational principle for free energy is reviewed. A systematic discussion is given of the approximate free energies of complex statistical systems. The analysis is centered around the variational principle of N. N. Bogoliubov for free energy in the context of its applications to various problems of statistical mechanics and condensed matter physics. The review presents a terse discussion of selected works carried out over the past few decades on the theory of many-particle interacting systems in terms of the variational inequalities. It is the purpose of this paper to discuss some of the general principles which form the mathematical background to this approach, and to establish a connection of the variational technique with other methods, such as the method of the mean (or self-consistent) field in the many-body problem, in which the effect of all the other particles on any given particle is approximated by a single averaged effect, thus reducing a many-body problem to a single-body problem. The method is illustrated by applying it to various systems of many-particle interacting systems, such as Ising and Heisenberg models, superconducting and superfluid systems, strongly correlated systems, etc. It seems likely that these technical advances in the many-body problem will be useful in suggesting new methods for treating and understanding many-particle interacting systems. This work proposes a new, general and pedagogical presentation, intended both for those who are interested in basic aspects, and for those who are interested in concrete applications.Comment: 60 pages, Refs.25

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

Optimization and Equilibrium Problems with Equilibrium Constraints

The paper concerns optimization and equilibrium problems with the so-called equilibrium constraints (MPEC and EPEC), which frequently appear in applications to operations research. These classes of problems can be naturally unified in the framework of multiobjective optimization with constraints governed by parametric variational systems (generalized equations, variational inequalities, complementarity problems, etc.). We focus on necessary conditions for optimal solutions to MPECs and EPECs under general assumptions in finite-dimensional spaces. Since such problems are intrinsically nonsmooth, we use advanced tools of generalized differentiation to study optimal solutions by methods of modern variational analysis. The general results obtained are concretized for special classes of MPECs and EPECs important in applications

Stability analysis for parameterized variational systems with implicit constraints

In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples. © 2019, The Author(s)

Quantitative Stability of Linear Infinite Inequality Systems under Block Perturbations with Applications to Convex Systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set $J$. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is $l_{\infty}(J)$. Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of [3] developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case

Quantitative Stability and Optimality Conditions in Convex Semi-Infinite and Infinite Programming

This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T . Parameter perturbations on the right-hand side of the inequalities are measurable and bounded, and thus the natural parameter space is $l_{\infty}(T)$. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map, which involves only the system data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. On one hand, in this way we extend to the convex setting the results of [4] developed in the linear framework under the boundedness assumption on the system coefficients. On the other hand, in the case when the decision space is reflexive, we succeed to remove this boundedness assumption in the general convex case, establishing therefore results new even for linear infinite and semi-infinite systems. The last part of the paper provides verifiable necessary optimality conditions for infinite and semi-infinite programs with convex inequality constraints and general nonsmooth and nonconvex objectives. In this way we extend the corresponding results of [5] obtained for programs with linear infinite inequality constraints