Uncertainty Quantification (UQ) is an emerging and extremely active research discipline which aims to quantitatively treat any uncertainty in applied models. The primary objective of Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and Applications is to present a comprehensive treatment of UQ in variational inequalities and some of its generalizations emerging from various network, economic, and engineering models. Some of the developed techniques also apply to machine learning, neural networks, and related fields.
- First book on UQ in variational inequalities emerging from various network, economic, and engineering models
- Completely self-contained and lucid in style
- Aimed for a diverse audience including applied mathematicians, engineers, economists, and professionals from academia
- Includes the most recent developments on the subject which so far have only been available in the research literature
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.