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Complementarity, equilibrium, efficiency and economics. (English) Zbl 1081.90001
Nonconvex Optimization and Its Applications 63. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0688-8/hbk). xvii, 445 p. (2002).
Several problems in so-called mathematical economics lead to complementarity problems in different degree of generality. Complementarity problems may sometimes be equivalently reduced to fixed point problems, variational inequalities, or other problems involving topological methods, variational methods, combinatorical methods, or methods from set-valued analysis. A classical source on the interconnections between these methods is the fine booklet by Kim C. Border [Fixed point theorems with applications to economics and game theory. Cambridge etc.: Cambridge University Press (1985; Zbl 0558.47038)].
In view of the increasing “formalization” of economical sciences (or rather pseudo-sciences), it is not surprising that there exists now a large number of monographs on complementarity and related problems and their application to economical equilibria. Classical examples from the last 20 years are the books by Ki Hang Kim and F. W. Roush [Competitive economics: equilibrium and arbitration. Amsterdam etc.: North-Holland (1983; Zbl 0524.90001)], Franklin M. Fisher [Disequilibrium foundations of equilibrium economics. Cambridge etc.: Cambridge University Press (1983; Zbl 0544.90001)], K. G. Murthy [Linear complementarity, linear and nonlinear programming. Berlin: Heldermann (1988; Zbl 0634.90037)], Richard W. Cottle et al. [The linear complementarity problem. Boston etc.: Academic Press (1992; Zbl 0757.90078)], Peter B. Dixon et al. [Notes and problems in applied general equilibrium economics. Amsterdam: North-Holland (1992; Zbl 0837.90017)], and È. V. Avtukhovich [Existence of a solution of the complementarity problem for multivalued mappings. Soobshcheniya po Prikladnoj Matematike. Moskva: Vychislitel’nyj Tsentr RAN (1999; Zbl 0935.90021)]. The first author of the book under review is also well-known for his Lecture Notes on ‘Complementarity problems’ [Lecture Notes in Mathematics. 1528. Berlin: Springer (1992; Zbl 0795.90072)] and a more recent monograph on ‘Topological methods in complementarity theory’ [Dordrecht: Kluwer Academic Publishers (2000; Zbl 0954.90056)].
In particular, the first author of this book has developed, together with his coauthors, the concept of so-called exceptional sequences and (more generally) exceptional families which constitute one useful approach to existence results in complementarity theory. So it is not surprising that this is one of the main topics covered in this book. To give an idea of the contents of its 13 chapters, divided into 4 parts, let us follow the authors’ description from the Preface.
Part I consists of 5 chapters. The first three chapters are dedicated to optimization models and general economic equilibrium. Chapter 1 deals with variational inequalities defined by set-valued operators, and the equivalence of the latter to fixed-point problems is shown. The notion of an exeptional family of points is introduced in a very general form, and a theorem of alternative of existence either of solutions or an exceptional family of elements, is proved. These results serve as tools for investigations presented in Parts II and III of the book. Chapter 2 is dedicated to optimization models involving single- and multivalued functions whose differential mapping is strictly quasi-monotone. Mathematical programs with feedback between the primal and dual variables, as well as multicriterial problems constitute a considerable part of this chapter. In Chapter 3, models of decentralized economy with competitive equilibrium and Arrow-Debreu models are presented, and theorems of existence of equilibria are proved. Chapter 4 is dedicated to generalized models of a single-product market. These models are based upon the conjectural equilibria and propose the whole range of intermediate models between the models of perfect competition, on the one hand, and the Cournot and Stackelberg models, on the other. Tools from complementarity theory are used to obtain conditions guaranteeing existence and uniqueness of equilibria in these generalized models. Chapter 5 scrutinizes in more detail the generalized Stackelberg model, allowing several leaders, instead of a single leader in the classical Stackelberg model. Existence of stationary points in thus extended model is established.
Part II that includes Chapter 6 and 7 applies the general notion of an exceptional family of elements to various particular classes of complementarity problems having obtained both well-known and new sufficient conditions of solvability of the complementarity problems. The majority of these results have already been published in previous papers and books by all three authors, but this is the first time that they all have been collected together. However, some results on solvability of Implicit Complementarity Problem presented in the last sections of Chapter 7 have not been yet published in papers.
Part III of the book is dedicated to description and properties of some numerical methods solving both linear and nonlinear complementarity problems. Chapter 8 presents various versions of the Lemke algorithm and its extensions by M. J. Todd and M. M. Kostreva. Chapter 9 deals with Scarf type algorithms solving linear complementarity problems, and follows mainly the work by P. J. J. Herings, A. A. J. Talman and Z. Yang [Math. Oper. Res. 21, No. 3, 675–696 (1996; Zbl 0870.90029)]. A Newton-like method for solving nonlinear complementarity problems and thus finding an equilibrium in the generalized Cournot model is examined in Chapter 10. Chapter 11 contains results of two types. First, some results on the parametrization approach to solving the nonlinear complementarity problem proposed by L. McLinden [Proc. Int. School Math., Erice/Sicily 1978, 251–270 (1980; Zbl 0499.90073)], is examined, a numerical method is presented and its convergence is proved. Second, quite promising algorithms using the Fischer-Burmeister merit functions developed in a series of recent papers by F. Facchinei, M. C. Ferris, M. S. Gowda, and C. Kanzow [e.g. Comput. Optim. Appl. 11, No. 3, 227–251 (1998; Zbl 0913.90250), Numer. Math. 80, No. 4, 557-577 (1998; Zbl 0915.65067), SIAM J. Optim. 9, No. 2, 342–373 (1999; Zbl 0986.90065), and SIAM J. Control Optim. 37, No. 4, 1150–1161 (1999; Zbl 0997.90085)] are also described.
Part IV consists of 2 chapters and is dedicated to efficiency problems. In Chapter 12, the authors present the most important notions of efficiency and study the relations between these notions. They establish several existence theorems related to efficiency and examine some properties of solution’s sets. Some Arrow-Barankin-Blackwell type theorems are also presented. The notion of nuclear cone is applied to the study of the efficiency. Finally, several topological properties of efficient point sets and a constructive test for Pareto efficiency are presented. Chapter 13 is dedicated to the study of several notions of approximative efficiency. The authors discuss the most important properties of each kind of approximative efficiency, and consider also the efficiency in product spaces and some interesting relations with Ekeland’s variational principle.
The presentation and style of the book seem rather inhomogeneous. In fact, Chapters 1–3 have been prepared by the second author, Chapters 4 and 5 by the third author, and Chapters 12 and 13 by the first author. As the authors write themselves, most of the material presented in this book has been published before, and thus should be known to the specialists in the field. On the other hand, non-specialists who want to learn something about complementarity problems, efficiency concepts, and economic equilibria will find a book which is merely the union of a large amount of material put together more or less occasionally by three different authors, of very limited use. Moreover, the price for the book is very high.

MSC:
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
49J40 Variational inequalities
90C29 Multi-objective and goal programming
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C47 Minimax problems in mathematical programming
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
91A40 Other game-theoretic models
91B50 General equilibrium theory
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