Foundations of Network Optimization and Games 1st Edition by Terry Friesz, David Bernstein 1489975942 9781489975942

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ISBN-10 :  1489975942 

ISBN-13 :  9781489975942

Author:  Terry L. Friesz, David Bernstein 

Foundations of Network Optimization and Games by Terry L. Friesz and David Bernstein is a book intended for scholars from all disciplines working on deterministic nonlinear network models expressed as nonlinear programs and mathematical games, as well as for graduate students encountering nonlinear network models for the first time. The book purposely emphasizes model formulation, algorithm selection, and illustrative numerical examples rather than theoretical results on convergence and algorithm complexity. As such Foundations of Network Optimization and Games will be especially useful to individuals who are builders of so-called computable models meant for direct decision support or comprehensive numerical discovery of the properties of complex, networked systems.

Foundations of Network Optimization and Games 1st Table of contents:

1 Introduction
1.1 Fundamental Notions
1.2 Transportation Networks
1.2.1 Traffic Prediction
1.2.2 Static Network Design
1.2.3 Real-Time Traffic Control
1.3 Telecommunication Networks
1.3.1 Quasi-static Flow Routing
1.3.2 Flow Control
1.4 Electric Power Networks
1.4.1 Modeling the Firms Generating Power
1.4.2 Modeling the ISO
1.4.3 The Complete Electric Power Model
1.5 Water Resource Networks
1.5.1 Irrigation Network Capacity Expansion Planning
1.5.2 Municipal Water Supply
1.6 The Way Ahead
1.7 References and Additional Reading
2 Elements of Nonlinear Programming
2.1 Nonlinear Program Defined
2.2 Other Types of Mathematical Programs
2.3 Necessary Conditions for an Unconstrained Minimum
2.4 Necessary Conditions for a Constrained Minimum
2.4.1 The Fritz John Conditions
2.4.2 Geometry of the Kuhn-Tucker Conditions
2.4.3 The Lagrange Multiplier Rule
2.4.4 Motivating the Kuhn-Tucker Conditions
2.5 Formal Derivation of the Kuhn-Tucker Conditions
2.5.1 Cones and Optimality
2.5.2 Theorems of the Alternative
2.5.3 The Fritz John Conditions Again
2.5.4 The Kuhn-Tucker Conditions Again
2.6 Sufficiency, Convexity, and Uniqueness
2.6.1 Quadratic Forms
2.6.2 Concave and Convex Functions
2.6.3 Kuhn-Tucker Sufficient Conditions
2.7 Generalized Convexity and Sufficiency
2.8 Sensitivity Analysis
2.9 Numerical and Graphical Examples
2.9.1 LP Graphical Solution
2.9.2 NLP Graphical Example
2.9.3 Nonconvex, Nongraphical Example
2.9.4 A Convex, Nongraphical Example
2.10 One-Dimensional Optimization
2.10.1 Derivative-Free Methods
2.10.2 Bolzano Search
2.10.3 Newton’s Method
2.10.4 Discretized Newton Methods
2.11 Descent Algorithms in n
2.12 References and Additional Reading
3 Elements of Graph Theory
3.1 Terms from Graph Theory
3.2 Network Notation
3.2.1 Single Copy Networks
3.2.2 Multicopy Networks
3.2.3 Remarks on Algorithm Complexity
3.3 Network Structure
3.3.1 The Nonlinear Minimum Cost Flow Problem
3.3.2 Linear Example of Network Structure
3.3.3 Formal Definitions
3.3.4 Implications of Network Structure
Problems with Natural Integer Solutions
Total Unimodularity
3.3.5 Near Network Structure
3.4 Labeling Algorithms
3.4.1 Path Generation and Tree Growing Problems
3.4.2 Path Generation and Tree Building
Example of Path Generation/Tree Building
3.4.3 Spanning Trees
Labeling with Multiple Arc Introduction1pt
Example of Multiple Arc Introduction Spanning Tree Algorithm
Single Node/Arc Introduction1pt
Simple Example1pt1pt1pt of MST Algorithm
3.4.4 The Minimum Path Problem
Finiteness and Complexity of Dijkstra’s Algorithm
Example of Dijkstra’s Algorithm1pt
3.4.5 Maximal Flow
A Path Augmentation Algorithm for the MFP
History, Finiteness and Complexity of Flow Augmentation Algorithm for the MFP
Example of Flow Augmentation Algorithm for the MFP
3.5 Solving the Linear Minimum Cost Flow Problem UsingGraph-Theoretic Methods
3.6 Hamiltonian Walks and the Traveling Salesman Problem
Preliminary Assumptions, Definitions and NP-Completeness
Algorithms
3.7 Summary
3.8 References and Additional Reading
4 Programs with Network Structure
4.1 The Revised Form of the Simplex
4.2 The Network Simplex
4.2.1 Bases for Network Programs
4.2.2 Determining the Entering Column
4.2.3 Determining the Exiting Column
4.2.4 The Network Simplex Algorithm and Initial Solutions
4.3 Degeneracy
4.4 Explicit Upper and Lower Bound Constraints
4.5 Detailed Example of the Network Simplex
4.6 Nonlinear Programs with Network Structure
4.6.1 Congestion
4.6.2 Feasible Direction Algorithms for Linearly Constrained NLPs
4.7 The Frank-Wolfe Algorithm
4.8 Steepest Descent Algorithm
4.9 A Primal Affine Scaling Algorithm
4.10 Nonlinear Network Example of the Frank-Wolfe Algorithm
4.11 Nonlinear Network Example of Primal Affine Scaling
4.12 Linear Network Example of Affine Scaling
4.13 References and Additional Reading
5 Near-Network and Large-Scale Programs
5.1 Programs with Near-Network Structure
5.2 Near-Network Examples
5.2.1 Time Constrained Shortest Paths
5.2.2 Minimum Cost Flow with Bundle Constraints
5.2.3 Multicommodity Flow
5.2.4 Congested Multicommodity Flow
5.2.5 Fuel Constrained Congested Flow
5.2.6 Common Structure
5.3 Nonlinear Programming Duality Theory
5.3.1 Definition of the Primal and Dual Programs
5.3.2 Global Optimality Conditions
5.3.3 Properties of the Dual Program
5.3.4 Comment on the Duality Gap
5.3.5 A Non-network Closed Form Example of Lagrangean Duality
5.3.6 Lagrangean Relaxation
5.3.7 Nondifferentiable and Subgradient Optimization
Subgradients and Their Properties
Ascent Algorithms for Nondifferentiable Problems
Subgradient Optimization Algorithm
5.4 A Non-network Example of Subgradient Optimization
5.5 Large-Scale Programs
5.5.1 Price Directive Versus Resource Directive Decomposition
5.5.2 Lagrangean Relaxation
5.6 The Representation Theorem
5.7 Dantzig-Wolfe Decomposition and Column Generation
5.8 Benders Decomposition
5.9 Simplicial Decomposition
5.10 References and Additional Reading
6 Normative Network Models and Their Solution
6.1 The Classical Linear Network Design Problem
6.2 The Transportation Problem
6.3 Variants of the Minimum Cost Flow Problem
6.3.1 Bundle Constraints and Lagrangean Relaxation
6.3.2 Bundle Constraints and Dantzig-Wolfe Decomposition
6.3.3 Nonlinear Unit Arc Costs
6.4 The Traveling Salesman Problem
6.5 The Vehicle Routing Problem
6.6 The Capacitated Plant Location Problem
6.7 Irrigation Networks
6.7.1 Irrigation Network Capacity Expansion Planning
6.7.2 Municipal Water Supply
6.7.3 Numerical Example
6.8 Telecommunications Flow Routing and System Optimal Traffic Assignment
6.9 References and Additional Reading
7 Nash Games
7.1 Some Basic Notions
7.2 Nash Equilibria and Normal Form Games
7.3 Variational Inequalities and Related Nonextremal Problems
7.4 Relationship of Variational Inequalities and MathematicalPrograms
7.5 Kuhn-Tucker Conditions for Variational Inequalities
7.6 Quasivariational Inequalities
7.7 Relationships Among Nonextremal Problems
7.8 Variational Inequality Representation of Nash Equilibrium
7.9 User Equilibrium
7.10 Variational Inequality Existence and Uniqueness
7.11 Sensitivity Analysis of Variational Inequalities
7.12 Diagonalization Algorithms
7.12.1 The Diagonalization Algorithm
7.12.2 Convergence of Diagonalization
7.12.3 A Nonnetwork Example of Diagonalization
7.13 Gap Function Methods for VI( F,Λ)
7.13.1 Gap Function Defined
7.13.2 The Auslender Gap Function
7.13.3 Fukushima-Auchmuty Gap Functions
7.13.4 The D-Gap Function
7.13.5 The D-Gap Function Algorithm
7.13.6 Convergence and Numerical Example of the D-Gap Algorithm
7.14 Other Algorithms for VI( F,Λ)
7.14.1 Methods Based on Differential Equations
7.14.2 Fixed Point Methods
7.14.3 Generalized Linear Methods
7.14.4 Successive Linearization and Lemke’s Algorithm
7.14.5 The Linear Complementarity Problem
Finding an Initial Feasible Solution for LCP(M,q)
Details of Lemke’s Method
Finiteness of Lemke’s Method
7.15 Computing Network User Equilibria
7.16 References and Additional Reading
8 Network Traffic Assignment
8.1 A Comment on Notation
8.2 System Optimal Traffic Assignment
8.2.1 Notation and Formulation
8.2.2 The Notion of Path-Based Network Structure
8.2.3 Arc Unit Latency Functions
8.2.4 Analysis of Necessary Conditions and Their Economic Interpretation
8.2.5 The Frank-Wolfe Algorithm for System Optimal Traffic Assignment
8.2.6 Variational Inequality Formulation of System Optimal Traffic Assignment
8.2.7 Existence of System Optimal Solutions and Sensitivity Analysis
8.2.8 Telecommunications Combined Routing and Flow Control Problem
8.2.9 Numerical Example: F-W Algorithm and System Optimal Traffic Assignment
8.3 User Optimal Traffic Assignment with Separable Functions
8.3.1 Uniqueness of User Equilibrium with Separable Functions
8.3.2 The Problem of Symmetry in Equivalent Optimization Problems
8.4 More About Nonseparable User Equilibrium
8.5 Frank-Wolfe Algorithm for Beckmann’s Program
8.6 Nonextremal Formulations of Wardropian Equilibrium
8.6.1 Asymmetric User Equilibrium with Fixed Demand
8.6.2 Asymmetric User Equilibrium with Elastic Demand
8.7 Diagonalization Algorithms for Nonextremal UserEquilibrium Models
8.8 Nonlinear Complementarity Formulation of UserEquilibrium
8.9 Numerical Examples of Computing User Equilibria
8.9.1 Frank-Wolfe Algorithm for Fixed Demand, Separable User Equilibrium
8.9.2 Diagonalization Applied to a Nonseparable User Equilibrium Problem
8.10 Sensitivity Analysis of User Equilibrium
8.10.1 Regularity and Formulae
8.10.2 Numerical Example
8.10.3 Cautionary Remarks
8.11 References and Additional Reading
9 Spatial Price Equilibrium on Networks
9.1 Extensions of STJ Network Spatial Price Equilibrium
9.1.1 Variational Inequality Formulation
9.1.2 Uniqueness
9.1.3 Existence
9.1.4 Equivalent Mathematical Program
9.2 Algorithms for STJ Network Spatial Price Equilibrium
9.2.1 Diagonalization Algorithms
9.2.2 Feasible Direction Algorithms
9.2.3 Numerical Example
9.3 Sensitivity Analysis for STJ Network Spatial PriceEquilibrium
9.3.1 Perturbation Analysis
9.3.2 Efficient Computation of Partial Derivatives with Respect to Perturbations
9.3.3 A Numerical Example
9.4 Oligopolistic Network Competition
9.5 Inclusion of Arbitrageurs
9.6 Modeling Freight Networks
9.6.1 Model Description
Shippers’ Submodel
Decomposition Algorithm
Carriers’ Submodel
Other Considerations
9.6.2 Model Validation
9.7 References and Additional Reading
10 Network Stackelberg Games and Mathematical Programs with Equilibrium Constraints
10.1 Defining the Price of Anarchy
10.2 Bounding the Price of Anarchy
10.2.1 Linear, Separable Arc Costs
10.2.2 When a Potential Function Exists
10.2.3 Importance of Bounding the Price of Anarchy
10.3 The Braess Paradox and Equilibrium Network Design
10.4 MPECs and Their Relationship to Stackelberg Games
10.5 Alternative Formulations of Network Equilibrium Design
10.5.1 The Discrete Equilibrium Network Design Model
10.5.2 The Continuous Equilibrium Network Design Problem
10.5.3 User Equilibrium Constraints
10.5.4 The Consumers’ Surplus Line Integral
10.5.5 Generating Paths
10.6 Algorithms for Continuous Equilibrium Network Design
10.6.1 Simulated Annealing
10.6.2 The Equilibrium Decomposed Optimization Heuristic
10.6.3 Computing When a Complementarity Formulation Is Employed
10.6.4 Computing When a Gap Function Is Employed
10.6.5 An Algorithm Based on Sensitivity Analysis
10.7 Numerical Comparison of Algorithms
10.8 Electric Power Markets
10.8.1 Modeling the Firms That Generate Power
10.8.2 Modeling the ISO
10.8.3 The Generalized Nash Game Among the Producers
10.8.4 The Stackelberg Game with the ISO as Leader
10.8.5 Solved Numerical Example of the MPEC Formulation
Description of the Network and Choice of Parameters
10.8.6 A PSO for MPEC
10.9 References and Additional Reading

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Tags: Foundations, Network Optimization, Games, Terry Friesz, David Bernstein