Sensitivity analysis for variational inequalities

Includes bibliographical references (leaves 25-26).Supported by the National Science Foundation Program in Systems Theory and Operations Research ECS-8316224by Yuping Qiu and Thomas L. Magnanti

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

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 $H^1$. 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

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.simulation;mathematical programs with equilibrium constraints;stability;regularity conditions;sample-path methods;stochastic mathematical programs with complementarity constraints