Uncertainty Quantification in Variational Inequalities

Joachim Gwinner is a retired Professor at the University of the Federal Army Munich. He earned his Ph.D. from University Mannheim in 1978. Then he was with Daimler-Benz company at Stuttgart for six years. After that, he became an Assistant Professor at Technical University Darmstadt and earned his Habilitation in 1989. His research interests lie in nonlinear and variational analysis, numerical analysis of partial differential equations, optimization theory and methods, and applications in continuum mechanics. He is the co-author of the monograph Advanced Boundary Element Methods: Treatment of Boundary Value, Transmission and Contact Problems. Baasansuren Jadamba earned her Ph.D. in Applied Mathematics and Scientific Computing from Friedrich-Alexander University Erlangen-Nuremberg (Germany) in 2004, and she is an Associate Professor at the School of Mathematical Sciences at the Rochester Institute of Technology. Her research interests and publications are in the numerical analysis of partial differential equations, finite element methods, parameter identification in partial differential equations, and stochastic equilibrium problems. Akhtar A. Khan is a Professor at the Rochester Institute of Technology. His research interests include inverse problems, optimal control, variational inequalities, and set-valued optimization. He is a co-author of the monograph Set-valued Optimization: An Introduction with Applications, Springer (2015) and co-editor of Nonlinear Analysis and Variational Problems: In Honor of George Isac, Springer (2009). Fabio Raciti earned his Ph.D. in Theoretical Physics from the University of Catania (Italy), where he has been an Assistant Professor and then an Associate Professor of Mathematical Analysis. He is currently an Associate Professor of Operations Research at the University of Catania and has received the National (Italian) Habilitation as a Full Professor of Operations Research. He has published research work in the field of variational inequalities, optimization, inverse problems, and stochastic equilibrium problems.

I. Variational Inequalities. 1. Preliminaries. 1.1. Elements of Functional Analysis. 1.2. Fundamentals of Measure Theory and Integration. 1.3. Essentials of Operator Theory. 1.4. An Overview of Convex Analysis and Optimization. 1.5. Comments and Bibliographical Notes. 2. Probability. 2.1. Probability Measure. 2.2. Conditional Probability and Independence. 2.3. Random Variables and Expectation. 2.4. Correlation, Independence, and Conditional Expectation. 2.5. Modes of Convergence of Random Variables. 2.6. Comments and Bibliographical Notes. 3. Projections on Convex Sets. 3.1. Projections on Convex Sets in Hilbert Spaces. 3.2. Projections on Convex Sets in Banach Spaces. 3.3. Comments and Bibliographical Notes. 4. Variational and Quasi-Variational Inequalities. 4.1. Illustrative Examples. 4.2. Linear Variational Inequalities. 4.3. Nonlinear Variational Inequalities. 4.4. Quasi Variational Inequalities. 4.5. Comments and Bibliographical Notes. 5. Numerical Methods for Variational and Quasi-Variational Inequalities. 5.1. Projection Methods. 5.2. Extragradient Methods. 5.3. Gap Functions and Descent Methods. 5.4. The Auxiliary Problem Principle. 5.5. Relaxation Method for Variational Inequalities. 5.6. Projection Methods for Quasi-Variational Inequalities. 5.7. Convergence of Recursive Sequences. 5.8. Comments and Bibliographical Notes. II. Uncertainty Quantification. Prologue on Uncertainty Quantification. 6. An Lp Approach for Variational Inequalities with Uncertain Data. 6.1. Linear Variational Inequalities with Random Data. 6.2. Nonlinear Variational Inequalities with Random Data. 6.3. Regularization of Variational Inequalities with Random Data. 6.4. Variational Inequalities with Mean-value Constraints. 6.5. Comments and Bibliographical Notes. 7. Expected Residual Minimization. 7.1. ERM for Linear Complementarity Problems. 7.2. ERM for Nonlinear Complementarity Problems. 7.3. ERM for Variational Inequalities. 7.4. Comments and Bibliographical Notes. 8. Stochastic Approximation Approach. 8.1. Stochastic Approximation. An Overview. 8.2. Gradient and Subgradient Stochastic Approximation. 8.3. Stochastic Approximation for Variational Inequalities. 8.4. Stochastic Iterative Regularization. 8.5. Stochastic Extragradient Method. 8.6. Incremental Projection Method. 8.7. Comments and Bibliographical Notes. III. Applications. 9. Uncertainty Quantification in Electric Power Markets. 9.1. Introduction. 9.2. The Model. 9.3. Complete Supply Chain Equilibrium Conditions. 9.4. Numerical Experiments. 9.5. Comments and Bibliographical Notes. 10. Uncertainty Quantification in Migration Models. 10.1. Introduction. 10.2. A Simple Model of Population Distributions. 10.3. A More Refined Model. 10.4. Numerical Examples. 10.5. Comments and Bibliographical Notes. 11. Uncertainty Quantification in Nash Equilibrium Problems. 11.1. Introduction. 11.2. Stochastic Nash Games and Variational Inequalities. 11.3. The Stochastic Oligopoly Model. 11.4. Uncertainty Quantification in Utility Functions. 11.5. Comments and Bibliographical Notes. 12. Uncertainty Quantification in Traffic Equilibrium Problems. 12.1 Introduction. 12.2. Traffic Equilibrium Problems via Variational Inequalities. 12.3. Uncertain Traffic Equilibrium Problems. 12.4. Computational Results. 12.5. A Comparative Study of Various Approaches. 12.6. Comments and Bibliographical Notes. Epilogue. Bibliography. Index.