Optimization and Multiobjective Control of Time-Discrete Systems

Dynamic Networks and Multilayered Structures

By: Dmitrii Lozovanu

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Hardcover | 6 March 2009

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Richard Bellmann developed a theory of dynamic programming which is for many reasons still in the center of great interest. The authors present a new approach in the ?eld of the optimization and multi-objective control of time-discrete systems which is closely related to the work of Richard Bellmann. They develop their own concept and their extension to the optimization and multi-objective control of time-discrete systems as well as to dynamic networks and multilayered structures are very stimulating for further research. Di?erent perspectives of discrete control and optimal dynamic ?ow problems on networks are treated and characterized. Together with the algorithmic solutions a framework of multi-objective control problems is - rived. The conclusion with a real world example underlines the necessity and - portance of their theoretic framework. As they come back to the classical Bellmann concept of dynamic programming they stress and honor his basic concept without debase their own work. Multilayereddecisionprocessesaspartofthedesignandanalysisofcomplexsystems and networks will be essential in many ways and ?elds in the future.

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Published: 19th October 2010

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Multi-Objective Control of Time-Discrete Systems and Dynamic Games on Networks	p. 1
Problem Formulation	p. 1
Single-Objective Discrete Control Problem	p. 2
Multi-Objective Control Based on the Concept of Non-cooperative Games: Nash Equilibria	p. 4
Hierarchical Control and Stackelberg's Optimization Principle	p. 7
Multi-Objective Control Based on the Concept of Cooperative Games: Pareto Optima	p. 8
Stationary and Non-Stationary Control of Time-Discrete Systems	p. 10
Multi-Objective Control of Time-Discrete Systems with Infinite Time Horizon	p. 10
Alternate Players' Control Condition and Nash Equilibria for Dynamic Games in Positional Form	p. 11
Algorithms for Solving Single-Objective Control Problems on Networks	p. 15
Dynamic Programming Algorithms for Solving Optimal Control Problems on Networks	p. 15
An Extension of Dijkstra's Algorithm for Optimal Control Problems with a Free Number of Stages	p. 18
Multi-Objective Control and Non-Cooperative Games on Dynamic Networks	p. 22
The Problem of Determining the Optimal Stationary Strategies in a Dynamic c-Game	p. 22
The Problem of Determining the Optimal Non-Stationary Strategies in a Dynamic c-Game	p. 25
Main Results for Dynamic c-Games with Constant Costs of the Edges and Determining Optimal Stationary Strategies of the Players	p. 26
Computational Complexity of the Problem of Determining Optimal Stationary Strategies in a Dynamic c-Game	p. 45
Determining the Optimal Stationary Strategies for a Dynamic c-Game with Non-Constant Cost Functions on the Edges	p. 45
Determining Nash Equilibria for Non-Stationary Dynamic c-Game	p. 53
Time-Expanded Networks for Non-Stationary Dynamic c-Game and Their Main Properties	p. 53
Determining Nash Equilibria	p. 55
Application of the Dynamic c-Game for Studying and Solving Multi-Objective Control Problems	p. 57
Multi-Objective Control and Cooperative Games on Dynamic Networks	p. 58
Stationary Strategies on Networks and Pareto Solutions	p. 58
A Pareto Solution for the Problem with Non-Stationary Strategies on Networks	p. 59
Determining Pareto Solutions for Multi-Objective Control Problems on Networks	p. 60
Determining Pareto Stationary Strategies	p. 60
Pareto Solution for the Non-Stationary Case of the Problem	p. 65
Computational Complexity of the Stationary Case of the Problem and an Algorithm for its Solving on Acyclic Networks	p. 65
Determining Pareto Optima for Multi-Objective Control Problems	p. 66
Determining a Stackelberg Solution for Hierarchical Control Problems	p. 67
A Stackelberg Solution for Static Games	p. 68
Hierarchical Control on Networks and Determining Stackelberg Stationary Strategies	p. 69
An Algorithm for Determining Stackelberg Stationary Strategies on Acyclic Networks	p. 73
An Algorithm for Solving Hierarchical Control Problems	p. 78
Max-Min Control Problems and Solving Zero-Sum Games on Networks	p. 81
Discrete Control and Finite Antagonistic Dynamic Games	p. 81
Max-Min Control Problem with Infinite Time Horizon	p. 82
Zero-Sum Games on Networks and a Polynomial Time Algorithm for Max-Min Paths Problems	p. 83
Problem Formulation	p. 84
An Algorithm for Solving the Problem on Acyclic Networks	p. 86
Main Results for the Problem on an Arbitrary Network	p. 88
A Polynomial Time Algorithm for Determining Optimal Strategies of the Players in a Dynamic c-Game	p. 90
A Pseudo-Polynomial Time Algorithm for Solving a Dynamic c-Game	p. 95
A Polynomial Time Algorithm for Solving Acyclic l-Games on Networks	p. 101
Problem Formulation	p. 101
Main Properties of Optimal Strategies in Acyclic l-Games	p. 102
A Polynomial Time Algorithm for Finding the Value and the Optimal Strategies in an Acyclic l-Game	p. 103
Cyclic Games: Algorithms for Finding the Value and the Optimal Strategies of the Players	p. 105
Problem Formulation and Main Properties	p. 106
Determining the Best Response of the First Player for a Fixed Strategy of the Second Player	p. 107
Some Preliminary Results	p. 110
The Reduction of Cyclic Games to Ergodic Games	p. 111
A Polynomial Time Algorithm for Solving Ergodic Zero-Value Cyclic Games	p. 111
A Polynomial Time Algorithm for Solving Cyclic Games Based on the Reduction to Acyclic l-Games	p. 113
An Approach for Solving Cyclic Games Based on a Dichotomy Method and Solving Dynamic c-Games	p. 116
Cyclic Games with Random States' Transitions of the Dynamical System	p. 117
A Nash Equilibria Condition for Cyclic Games with p Players	p. 118
Determining Pareto Optima for Cyclic Games with p Players	p. 122
Extension and Generalization of Discrete Control Problems and Algorithmic Approaches for its Solving	p. 125
Discrete Control Problems with Varying Time of States' Transitions of the Dynamical System	p. 125
The Single-Objective Control Problem with Varying Time of States' Transitions of the Dynamical System	p. 126
An Algorithm for Solving a Single-Objective Control Problem with Varying Time of States' Transitions of the Dynamical System	p. 127
The Discrete Control Problem with Cost Functions of System's Passages that Depend on the Transition-Time of States' Transitions	p. 132
The Control Problem on a Network with Transition-Time Functions on the Edges	p. 133
Problem Formulation	p. 133
An Algorithm for Solving the Problem on a Network with Transition-Time Functions on the Edges	p. 134
Multi-Objective Control of Time-Discrete Systems with Varying Time of States' Transitions	p. 141
Multi-Objective Discrete Control with Varying Time of States' Transitions of Dynamical Systems	p. 141
A Dynamic c-Game on Networks with Transition-Time Functions on the Edges	p. 146
Remark on Determining Pareto Optima for the Multi-Objective Control Problem with Varying Time of States' Transitions	p. 149
An Algorithm for Solving the Discrete Optimal Control Problem with Infinite Time Horizon and Varying Time of the States' Transitions	p. 150
Problem Formulation and Some Preliminary Results	p. 150
An Algorithm for Determining an Optimal Stationary Control for Dynamical Systems with Infinite Time Horizon	p. 152
A General Approach for Algorithmic Solutions of Discrete Optimal Control Problems and its Game-Theoretic Extension	p. 154
A General Optimal Control Model	p. 154
An Algorithm for Determining an Optimal Solution of the Problem with Fixed Starting and Final States	p. 156
The Discrete Optimal Control Problem on a Network	p. 159
The Game-Theoretic Control Model with p Players	p. 160
The Game-Theoretic Control Problem on Networks and an Algorithm for its Solving	p. 161
Multi-Criteria Discrete Control Problems: Pareto Optima	p. 169
Pareto-Nash Equilibria for Multi-Objective Games	p. 171
Problem Formulation	p. 172
Main Results	p. 173
Discrete and Matrix Multi-Objective Games	p. 177
Some Comments on and Interpretations of Multi-Objective Games	p. 179
Determining a Pareto-Stackelberg Solution for Multi-Objective Games	p. 179
Discrete Control and Optimal Dynamic Flow Problems on Networks	p. 181
Single-Commodity Dynamic Flow Problems and the Time-Expanded Network Method for Their Solving	p. 181
The Minimum Cost Dynamic Flow Problem	p. 182
The Main Results	p. 183
The Dynamic Model with Flow Storage at Nodes	p. 186
The Dynamic Model with Flow Storage at Nodes and Integral Constant Demand-Supply Functions	p. 188
The Algorithm	p. 189
Constructing the Time-Expanded Network and its Size	p. 190
Approaches for Solving the Minimum Cost Flow Problem with Different Types of Cost Functions on the Edges	p. 200
Determining the Minimum Cost Flows in Dynamic Networks with Transition Time Functions that Depend on Flow and Time	p. 208
An Algorithm for Solving the Maximum Dynamic Flow Problem	p. 212
Multi-Commodity Dynamic Flow Problems and Algorithms for their Solving	p. 214
The Minimum Cost Multi-Commodity Dynamic Flow Problem	p. 214
The Main Results	p. 216
The Algorithm	p. 220
Examples	p. 220
The Dynamic Multi-Commodity Minimum Cost Flow Problem with Transition Time Functions that Depend on Flows and on Time	p. 224
Generalizations	p. 229
An Algorithm for Solving the Maximum Dynamic Multi-Commodity Flow Problem	p. 229
The Game-Theoretic Approach for Dynamic Flow Problems on Networks	p. 231
Applications and Related Topics	p. 233
Analysis and Control of Time-Discrete Systems: Resource Planning - The TEM Model	p. 233
Motivation	p. 234
The Basic Model	p. 234
Control Theoretic Part	p. 237
Problem of Fixed Point Controllability and Null-Controllability	p. 238
Optimal Investment Parameter	p. 240
A Game-Theoretic Extension - Relation to Multilayered Decision Problems	p. 244
Algorithmic Solutions for an Emission Reduction Game: The Kyoto Game	p. 250
The Core in the TEM Model	p. 250
A Second Cooperative Treatment of the TEM Model	p. 259
Comments	p. 268
An Emission Reduction Process - The MILAN Model	p. 269
MILAN: Multilayered Games on Networks The General Kyoto Game as a Multi-Step Process	p. 269
Sequencing and Dynamic Programming	p. 271
Generalizations of the Feasible Decision Sets: Optimal Solutions on k-Layered Graphs	p. 274
Conclusion	p. 275
References	p. 277
Index	p. 283
Table of Contents provided by Ingram. All Rights Reserved.

Optimization and Multiobjective Control of Time-Discrete Systems

Dynamic Networks and Multilayered Structures

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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Other Editions and Formats

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