Optimization and Multiobjective Control of Time-Discrete Systems: Dynamic Networks and Multilayered Structures

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Springer Science & Business Media, Apr 21, 2009 - Mathematics - 285 pages
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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Contents

MultiObjective Control of TimeDiscrete Systems and Dynamic Games on Networks
1
111 SingleObjective Discrete Control Problem
2
Nash Equilibria
4
113 Hierarchical Control and Stackelbergs Optimization Principle
7
Pareto Optima
8
115 Stationary and NonStationary Control of TimeDiscrete Systems
10
13 Alternate Players Control Condition and Nash Equilibria for Dynamic Games in Positional Form
11
14 Algorithms for Solving SingleObjective Control Problems on Networks
15
32 The Control Problem on a Network with TransitionTime Functions on the Edges
133
322 An Algorithm for Solving the Problem on a Network with TransitionTime Functions on the Edges
134
33 MultiObjective Control of TimeDiscrete Systems with Varying Time of States Transitions
141
332 A Dynamic cGame on Networks with TransitionTime Functions on the Edges
146
333 Remark on Determining Pareto Optima for the MultiObjective Control Problem with Varying Time of States Transitions
149
34 An Algorithm for Solving the Discrete Optimal Control Problem with Infinite Time Horizon and Varying Time of the States Transitions
150
342 An Algorithm for Determining an Optimal Stationary Control for Dynamical Systems with Infinite Time Horizon
152
35 A General Approach for Algorithmic Solutions of Discrete Optimal Control Problems and its GameTheoretic Extension
154

142 An Extension of Dijkstras Algorithm for Optimal Control Problems with a Free Number of Stages
18
15 MultiObjective Control and NonCooperative Games on Dynamic Networks
22
152 The Problem of Determining the Optimal NonStationary Strategies in a Dynamic cGame
25
16 Main Results for Dynamic cGames with Constant Costs of the Edges and Determining Optimal Stationary Strategies of the Players
26
17 Computational Complexity of the Problem of Determining Optimal Stationary Strategies in a Dynamic cGame
45
19 Determining Nash Equilibria for NonStationary Dynamic cGames
53
192 Determining Nash Equilibria
55
110 Application of the Dynamic cGame for Studying and Solving MultiObjective Control Problems
57
111 MultiObjective Control and Cooperative Games on Dynamic Networks
58
1112 A Pareto Solution for the Problem with NonStationary Strategies on Networks
59
112 Determining Pareto Solutions for MultiObjective Control Problems on Networks
60
1122 Pareto Solution for the NonStationary Case of the Problem
65
113 Determining Pareto Optima for MultiObjective Control Problems
66
114 Determining a Stackelberg Solution for Hierarchical Control Problems
67
1141 A Stackelberg Solution for Static Games
68
1142 Hierarchical Control on Networks and Determining Stackelberg Stationary Strategies
69
1143 An Algorithm for Determining Stackelberg Stationary Strategies on Acyclic Networks
73
1144 An Algorithm for Solving Hierarchical Control Problems
78
MaxMin Control Problems and Solving ZeroSum Games on Networks
80
22 MaxMin Control Problem with Infinite Time Horizon
82
23 ZeroSum Games on Networks and a Polynomial Time Algorithm for MaxMin Paths Problems
83
231 Problem Formulation
84
232 An Algorithm for Solving the Problem on Acyclic Networks
86
233 Main Results for the Problem on an Arbitrary Network
88
234 A Polynomial Time Algorithm for Determining Optimal Strategies of the Players in a Dynamic cGame
90
235 A PseudoPolynomial Time Algorithm for Solving a Dynamic cGame
95
24 A Polynomial Time Algorithm for Solving Acyclic lGames on Networks
101
242 Main Properties of Optimal Strategies in Acyclic lGames
102
243 A Polynomial Time Algorithm for Finding the Value and the Optimal Strategies in an Acyclic lGame
103
Algorithms for Finding the Value and the Optimal Strategies of the Players
105
251 Problem Formulation and Main Properties
106
252 Determining the Best Response of the First Player for a Fixed Strategy of the Second Player
107
253 Some Preliminary Results
110
254 The Reduction of Cyclic Games to Ergodic Games
111
256 A Polynomial Time Algorithm for Solving Cyclic Games Based on the Reduction to Acyclic lGames
113
257 An Approach for Solving Cyclic Games Based on a Dichotomy Method and Solving Dynamic cGames
116
26 Cyclic Games with Random States Transitions of the Dynamical System
117
27 A Nash Equilibria Condition for Cyclic Games with p Players
118
28 Determining Pareto Optima for Cyclic Games with p Players
122
Extension and Generalization of Discrete Control Problems and Algorithmic Approaches for its Solving
125
311 The SingleObjective Control Problem with Varying Time of States Transitions of the Dynamical System
126
312 An Algorithm for Solving a SingleObjective Control Problem with Varying Time of States Transitions of the Dynamical System
127
313 The Discrete Control Problem with Cost Functions of Systems Passages that Depend on the TransitionTime of States Transitions
132
352 An Algorithm for Determining an Optimal Solution of the Problem with Fixed Starting and Final States
156
353 The Discrete Optimal Control Problem on a Network
159
354 The GameTheoretic Control Model with p Players
160
355 The GameTheoretic Control Problem on Networks and an Algorithm for its Solving
161
Pareto Optima
169
36 ParetoNash Equilibria for MultiObjective Games
171
361 Problem Formulation
172
362 Main Results
173
363 Discrete and Matrix MultiObjective Games
177
364 Some Comments on and Interpretations of MultiObjective
179
Discrete Control and Optimal Dynamic Flow Problems on Networks
180
411 The Minimum Cost Dynamic Flow Problem
182
412 The Main Results
183
413 The Dynamic Model with Flow Storage at Nodes
186
414 The Dynamic Model with Flow Storage at Nodes and Integral Constant DemandSupply Functions
188
415 The Algorithm
189
416 Constructing the TimeExpanded Network and its Size
190
417 Approaches for Solving the Minimum Cost Flow Problem with Different Types of Cost Functions on the Edges
200
418 Determining the Minimum Cost Flows in Dynamic Networks with Transition Time Functions that Depend on Flow and Time
208
419 An Algorithm for Solving the Maximum Dynamic Flow Problem
212
42 MultiCommodity Dynamic Flow Problems and Algorithms for their Solving
214
422 The Main Results
216
423 The Algorithm
220
425 The Dynamic MultiCommodity Minimum Cost Flow Problem with Transition Time Functions that Depend on Flows and on Time
224
426 Generalizations
229
43 The GameTheoretic Approach for Dynamic Flow Problems on Networks
231
Applications and Related Topics
233
511 Motivation
234
513 Control Theoretic Part
237
514 Problem of Fixed Point Controllability and NullControllability
238
515 Optimal Investment Parameter
240
516 A GameTheoretic Extension Relation to Multilayered Decision Problems
244
The Kyoto Game
250
522 A Second Cooperative Treatment of the TEM Model
259
523 Comments
268
53 An Emission Reduction Process The MILAN Model
269
532 Sequencing and Dynamic Programming
271
Optimal Solutions on kLayered Graphs
274
Conclusion
275
References
277
Index
283
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