Dynamic optimization and differential games.

*(English)*Zbl 1207.91004
International Series in Operations Research & Management Science 135. New York, NY: Springer (ISBN 978-0-387-72777-6/hbk; 978-0-387-09496-0/ebook). xiv, 499 p. (2010).

This book is devoted to optimal control and differential game problems with emphasis on economic applications such as revenue management, oligopoly pricing, production planning, supply chain management, and dynamic network problems. The author describes (differential) variational inequalities as a basic tool for solving these problems.

The book contains a preface, ten chapters, and an index. Chapter 1 gives a brief introduction to the history of calculus of variations and optimal control supported by examples in optimal economic growth, allocation of public investment, and dynamic telecommunications. Then it briefly describes differential games and applications in dynamic vehicular and oligopolistic competition networks, revenue management and pricing. Chapter 2 first describes basics in finite-dimensional nonlinear programming and then their applications to discrete-time optimal control problems. A more detailed exposition of calculus of variations and continuous time optimal control with emphasis on optimality conditions as well as selected examples are given in Chapter 3. Chapter 4 contains elements of functional analysis, optimality conditions for infinite-dimensional optimization problems mostly in the form of variational inequalities, and some selected optimization methods such as steepest descent, projection gradient and penalty ones. Chapter 5 presents finite-dimensional variational inequalities and their relationship with complementarity, fixed-point, and Nash equilibrium problems. Besides, some results on sensitivity for variational inequalities and selected iterative methods such as diagonalization, gap function, fixed-point, and linearization ones are also included. Chapter 6 presents the foundations of dynamic Nash games without explicit consideration of feedback, their relation with differential variational inequalities, and, also, adjustment of fixed-point and gap function based algorithms for these problems. The next chapters are completely devoted to applications. Namely, Chapter 7 describes some models of optimal economic growth and a model of regional public investment allocation. Chapter 8 contains some dynamic perfect and imperfect competition models and their spatial extensions, including a dynamic oligopolistic spatial one and competitive supply chains. Chapter 9 describes applications to dynamic models of user equilibrium which are utilized in particular for prediction of flow distributions in road networks. Also, Chapter 10 describes some differential Nash games arising in network revenue management and their variational inequality type formulations.

The presentation is given at a student’s level, contains many significant examples and exercises, so that it can be used as a textbook.

However, there are some comments which can be regarded as possible suggestions for improvements of further editions. Firstly, it would be desirable to give a more detailed survey of differential game theory and applications; see e.g. [N. N. Krasovskii and A. I. Subbotin, Game-theoretical control problems. Springer Series in Soviet Mathematics. New York etc.: Springer-Verlag (1988; Zbl 0649.90101)], [E. M. Vaisbord and V. I. Zhukovskii, Introduction to multi-player differential games and their applications. New York: Gordon and Breach (1988)] and [L. A. Petrosyan, Differential games of pursuit. Series on Optimization. 2. Singapore: World Scientific (1993; Zbl 0799.90144)].

Secondly, it would be worth while to outline related results on dynamic network equilibrium problems; see e.g. [P. Daniele, F. Giannessi and A. Maugeri, Equilibrium problems and variational models. Nonconvex Optimization and Its Applications. 68. Boston, MA: Kluwer Academic Publishers (2003; Zbl 1030.00031)].

Next, the notion “nonextremal problem” used in Chapter 5 does not seem suitable, and can be replaced by e.g. “equilibrium type problem”. The primal gap function (p. 249) was introduced by S. I. Zukhovitskij, R. A. Polyak and M. E. Primak [Sov. Math., Dokl. 10, 279–282 (1969; Zbl 0191.49801)]. When describing gap function properties in Chapters 5–6, it would be worth while to add conditions which provide equivalence between the variational inequality and stationary point problem.

The book contains a preface, ten chapters, and an index. Chapter 1 gives a brief introduction to the history of calculus of variations and optimal control supported by examples in optimal economic growth, allocation of public investment, and dynamic telecommunications. Then it briefly describes differential games and applications in dynamic vehicular and oligopolistic competition networks, revenue management and pricing. Chapter 2 first describes basics in finite-dimensional nonlinear programming and then their applications to discrete-time optimal control problems. A more detailed exposition of calculus of variations and continuous time optimal control with emphasis on optimality conditions as well as selected examples are given in Chapter 3. Chapter 4 contains elements of functional analysis, optimality conditions for infinite-dimensional optimization problems mostly in the form of variational inequalities, and some selected optimization methods such as steepest descent, projection gradient and penalty ones. Chapter 5 presents finite-dimensional variational inequalities and their relationship with complementarity, fixed-point, and Nash equilibrium problems. Besides, some results on sensitivity for variational inequalities and selected iterative methods such as diagonalization, gap function, fixed-point, and linearization ones are also included. Chapter 6 presents the foundations of dynamic Nash games without explicit consideration of feedback, their relation with differential variational inequalities, and, also, adjustment of fixed-point and gap function based algorithms for these problems. The next chapters are completely devoted to applications. Namely, Chapter 7 describes some models of optimal economic growth and a model of regional public investment allocation. Chapter 8 contains some dynamic perfect and imperfect competition models and their spatial extensions, including a dynamic oligopolistic spatial one and competitive supply chains. Chapter 9 describes applications to dynamic models of user equilibrium which are utilized in particular for prediction of flow distributions in road networks. Also, Chapter 10 describes some differential Nash games arising in network revenue management and their variational inequality type formulations.

The presentation is given at a student’s level, contains many significant examples and exercises, so that it can be used as a textbook.

However, there are some comments which can be regarded as possible suggestions for improvements of further editions. Firstly, it would be desirable to give a more detailed survey of differential game theory and applications; see e.g. [N. N. Krasovskii and A. I. Subbotin, Game-theoretical control problems. Springer Series in Soviet Mathematics. New York etc.: Springer-Verlag (1988; Zbl 0649.90101)], [E. M. Vaisbord and V. I. Zhukovskii, Introduction to multi-player differential games and their applications. New York: Gordon and Breach (1988)] and [L. A. Petrosyan, Differential games of pursuit. Series on Optimization. 2. Singapore: World Scientific (1993; Zbl 0799.90144)].

Secondly, it would be worth while to outline related results on dynamic network equilibrium problems; see e.g. [P. Daniele, F. Giannessi and A. Maugeri, Equilibrium problems and variational models. Nonconvex Optimization and Its Applications. 68. Boston, MA: Kluwer Academic Publishers (2003; Zbl 1030.00031)].

Next, the notion “nonextremal problem” used in Chapter 5 does not seem suitable, and can be replaced by e.g. “equilibrium type problem”. The primal gap function (p. 249) was introduced by S. I. Zukhovitskij, R. A. Polyak and M. E. Primak [Sov. Math., Dokl. 10, 279–282 (1969; Zbl 0191.49801)]. When describing gap function properties in Chapters 5–6, it would be worth while to add conditions which provide equivalence between the variational inequality and stationary point problem.

Reviewer: Igor V. Konnov (Kazan)

##### MSC:

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

91A23 | Differential games (aspects of game theory) |

47J20 | Variational and other types of inequalities involving nonlinear operators (general) |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C39 | Dynamic programming |

91B55 | Economic dynamics |

90B10 | Deterministic network models in operations research |

91B72 | Spatial models in economics |