Nonlinear Programming Techniques for Equilibria

This book considers a range of problems in operations research, which are formulated through various mathematical models such as complementarity, variational inequalities, multiobjective optimization, fixed point problems, noncooperative games and inverse optimization. Moreover, the book subsumes all these models under a common structure that allows them to be formulated in a unique format: the Ky Fan inequality. It subsequently focuses on this unifying equilibrium format, providing a comprehensive overview of the main theoretical results and solution algorithms, together with a wealth of applications and numerical examples. Particular emphasis is placed on the role of nonlinear optimization techniques – e.g. convex optimization, nonsmooth calculus, proximal point and descent algorithms – as valuable tools for analyzing and solving Ky Fan inequalities. . Read more... Abstract: This book considers a range of problems in operations research, which are formulated through various mathematical models such as complementarity, variational inequalities, multiobjective optimization, fixed point problems, noncooperative games and inverse optimization. Moreover, the book subsumes all these models under a common structure that allows them to be formulated in a unique format: the Ky Fan inequality. It subsequently focuses on this unifying equilibrium format, providing a comprehensive overview of the main theoretical results and solution algorithms, together with a wealth of applications and numerical examples. Particular emphasis is placed on the role of nonlinear optimization techniques – e.g. convex optimization, nonsmooth calculus, proximal point and descent algorithms – as valuable tools for analyzing and solving Ky Fan inequalities Preface......Page 3 Contents......Page 5 Obstacle Problem......Page 7 Power Control in Wireless......Page 9 Traffic Network......Page 10 Portfolio Selection......Page 12 Optimal Production under Restricted Resources......Page 13 IO Analysis in Economy......Page 14 Quality Control in Production Systems......Page 16 Complementarity Problems......Page 17 Variational Inequality......Page 18 Weak Pareto Optimization......Page 19 Fixed Point......Page 20 Inverse Optimization......Page 21 Notes & Refs......Page 22 Convexity & Monotonicity......Page 23 Equivalent Reformulations......Page 30 Existence......Page 37 Stability......Page 45 Error Bounds......Page 49 Notes & Refs......Page 53 Fixed-Point & Extragradient Methods......Page 57 Descent Methods......Page 65 Gap Functions......Page 67 D-Gap Functions......Page 68 Regularization Methods......Page 71 Parameters' Calibration......Page 74 Notes & Refs......Page 75 Applications......Page 78 Traffic Networks & Quasi-Variational Inequalities......Page 79 Theory......Page 80 Fixed-Point & Extragradient Methods......Page 85 Descent Methods......Page 92 Notes & Refs......Page 101 Topological Concepts......Page 103 Functions......Page 107 Multivalued Maps......Page 112 Refs......Page 117 Index......Page 123