7,287 research outputs found
Parametric general quasi variational inequalities
In this paper, we prove that the general quasi variational inequalities are equivalent to the fixed point problem using the projection technique. We use this equivalent formulation to study the sensitivity analysis of the general quasi variational inequality. Our approach is very simple. Several special cases are also discussed
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
Shape optimisation for a class of semilinear variational inequalities with applications to damage models
The present contribution investigates shape optimisation problems for a class
of semilinear elliptic variational inequalities with Neumann boundary
conditions. Sensitivity estimates and material derivatives are firstly derived
in an abstract operator setting where the operators are defined on polyhedral
subsets of reflexive Banach spaces. The results are then refined for
variational inequalities arising from minimisation problems for certain convex
energy functionals considered over upper obstacle sets in . One
particularity is that we allow for dynamic obstacle functions which may arise
from another optimisation problems. We prove a strong convergence property for
the material derivative and establish state-shape derivatives under regularity
assumptions. Finally, as a concrete application from continuum mechanics, we
show how the dynamic obstacle case can be used to treat shape optimisation
problems for time-discretised brittle damage models for elastic solids. We
derive a necessary optimality system for optimal shapes whose state variables
approximate desired damage patterns and/or displacement fields
Some aspects of variational inequalities
AbstractIn this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence, uniqueness, computational properties, various generalizations, sensitivity analysis and their applications. We also propose some open problems with sufficient information and references, so that someone may attempt solution(s) in his/her area of special interest. We also include some new results, which we have recently obtained
Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints: Sample-Path Analysis
We consider a class of stochastic mathematical programs with complementarity constraints, in which both the objective and the constraints involve limit functions or expectations that need to be estimated or approximated.Such programs can be used for modeling average or steady-state behavior of complex stochastic systems.Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models.Here we broaden the applicability of so-called the sample-path method to include the solution of certain stochastic mathematical programs with equilibrium constraints.The convergence analysis of sample-path methods rely heavily on stability conditions.We first review necessary sensitivity results, then describe the method, and provide sufficient conditions for its almost-sure convergence.Alongside we provide a complementary sensitivity result for the corresponding deterministic problems.In addition, we also provide a unifying discussion on alternative set of sufficient conditions, derive a complementary result regarding the analysis of stochastic variational inequalities, and prove the equivalence of two different regularity conditions.stochastic processes;mathematics;stability;simulation;regulations;general equilibrium
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