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The goal is to elaborate on the main principles of the theory of the Berge equilibrium by answering the following two questions: What are the basic properties of the Berge equilibrium? Does the Berge equilibrium exist, and how can it be calculated? The approach presented here will be of particular interest to researchers (including undergraduates and graduates) and economists focused on decision-making under complex conflict conditions. The peaceful resolution of conflicts is the cornerstone of the approach: as a matter of fact, the Golden Rule precludes military clashes and violence. In turn, the new approach requires new methods; in particular, the existence problems are reduced to saddle point design for the Germeier convolution of payoff functions, with further transition to mixed strategies in accordance with the standard procedure employed by E. Borel, J. von Neumann, J. Nash, and their followers. 

This book synthesizes the game-theoretic modeling of decision-making processes and an ancient moral requirement called the Golden Rule of ethics (GR). This rule states "Behave to others as you would like them to behave to you." The GR is one of the oldest, most widespread, and specific moral requirements that appear in Christianity, Islam, Judaism, Buddhism, and Confucianism.

This book constructs and justifies mathematical models of dynamic socio-economic processes and phenomena that reveal the mechanism of the GR and are based on the concept of Berge equilibrium. The GR can be naturally used for resolving or balancing conflicts, and its "altruistic character" obviously excludes wars, blood-letting, and armed clashes.

This book will be of use to undergraduate and graduate students and experts in the field of decision-making in complex control and management systems, as well as anyone interested in game theory and applications.

The questions of the formalization of the estimates of the phenomenon state, which can be used in the decision-making multicriterion problems [ Vector- Valued Optimization Problems in Control Theory (Academic Press, New York, 1979)] taking into account the metrics of the space of partial criteria are considered. Any phenomenon is described by some system of characteristic parameters, which values are presented by the coordinates of points in the space of states of the given phenomenon. A notion of the "ideal" state determined by optimal values of each characteristic parameter separately and independently of others is used. Estimate of any state is determined by the distance between that state and the "ideal" one. That permits to choose from a multitude of the given states the best one, to which conforms a minimum distance to the "ideal" state. In the general case of the curved space the distances are measured by geodesic lines drawn by using a metric tensor of the given space. The metric tensor components themselves are determined by the solution of the differential equations set up from the condition of minimization of a certain functional. © World Scientific Publishing Company.

Structural and parametric identification of nonlinear continuous dynamic systems with a closed cycle on a set of continuous block-oriented models with feedback is considered. The method of structural identification in the steady state based on the observation of the system's input and output variables at the input periodic influences is proposed. The solution of the parameter identification problems, which can be immediately connected with the structural identification problem, is carried out in the steady and transient states by the method of least squares. The structural and parametric identification algorithms are investigated by means of both theoretical analysis and computer modeling. © 2013 World Scientific Publishing Company.

The problem of the construction of an object functioning in the regime of optimum performance at the design stage is reduced to the solution of the problem of multicriteria optimization, where the quality criteria are chosen to be its most essential characteristics. At the same time in all methods of multicriterion optimization the vector quality criterion is considered basically in the linear Euclidean space. Actually, in most cases, the criterion space is non-Euclidean - it is curved. Therefore, such setting cannot give results adequately reflecting the processes running in real systems. In order for the design system to really satisfy the optimality requirements the authors of the given paper offer an absolutely new approach to the solution of the problems of multicriteria optimization based on the definition of the quality criteria space and on finding an invariant corresponding to the distance between any two points of that space. The idea of the study of the metric properties of the quality criteria space and their use in solving problems of multicriteria optimization was offered in the work. When solving such problems the quality criteria space was automatically identified with the Euclidean space with corresponding metrics. In the general case this could not give results adequately reflecting the processes occurring in real systems. In the present paper metric properties of space criteria are studied for the first time, using as the main instrument the mathematical apparatus of tensor analysis, Riemannian geometry, differential equations in partial derivatives etc. Boundary problems relative to the components of the metric tensor of the n-dimensional space of the phenomenon states enabling to determine its metric properties are posed. The knowledge of the metric tensor furthers the objective appraisal of the phenomenon state and the definition of the optimal state.

This article is about the full analysis of one concrete class of general non-cooperative lexicographic games and its computer programming. In such game, the payoffs of players are lexicographic vector payoffs m scalar criteria vectors. At the same time, these criteria are strictly ranked on the set of the situations with lexicographic preference. In some such kind of game a Nash's equilibrium may not exist. In the given article the full analysis of one class of dyadic lexicographic games is worked out. Such kind of class is the non-cooperative lexicographic games, where each player has got two pure strategies and the payoff of each player solely depends on the strategies of two players in each situation. Therefore, the player's payoffs are given by 2 × 2 matrices, the elements of which are lexicographic utilities. © 2009 World Scientific Publishing Company.

The basis of the safe operation of ecologically dangerous objects is formed at the design stage. Therefore, the primary goal of this stage is, whenever possible, full execution of technical and safety requirements that provide reliable safety systems and their effective maintenance. Among ecologically dangerous objects, nuclear power stations (NPSs) are especially important, since their normal operation is a basic problem. Multicriteria design problems and maintenance strategies of NPS protection systems are investigated in this paper. Using the example of an NPS, three optimization models for effective design and operation of ecologically dangerous objects are developed. In particular the model for effective maintenance of the NPS protection system is constructed. The model of constructing the NPS reliable protection system with minimal expenditures with primitive constraints and generalized ones is designed as well. We reduce these problems to multiobjective optimization (MO) problems. We have developed our method, the appropriate algorithm and the corresponding software for solving MO problems based on the gravitation centers method. Our software yields solutions for MO problems of effective design and maintenance service of a NPS, minimizing the risk of abnormal functioning and providing design of a reliable system of protection at minimal expense.

Multicriteria (multiobjective) optimization is widely used for solving a variety of problems in all spheres of everyday life. This could be explained by the fact that such models make it possible to take into account goals which may be different in nature and very often quite contradictory Researchers have developed and continued to improve methods, algorithms and procedures for solving such problems. In this paper we investigate various multicriteria optimization methods and propose an approach for classification of these methods, so that a decision maker can choose from a great variety of procedures the one that best suits his/her goals. Further, we show how our gravitation centers method, developed earlier for solving single objective programs, can be implemented through the multicriteria or multiobjective case. 

Structural and parametric identification of the nonlinear systems with a closed cycle on the set of continuous block-oriented models with feedback is considered. The method of the structural identification in the steady state based on the observation of the system's input and output variables at the input periodic influences is proposed. The solution of the parameter identification problems, which can be immediately connected with the structural identification problem, is carried out in the steady and transient states by the method of least squares. The structural and parametric identification algorithms are investigated by means of both the theoretical analysis and the computer modeling.

The problem of parameter identification of linear non-stationary dynamical systems is considered. It is supposed that a direct observation of the state vector of system is not possible and elements of the output matrix are variables. Stated is the problem of parameter identification as, in a certain sense, an inverse problem of Cauchy’s problem for linear ordinary differential equations. Under some restrictions on the system’s parameters, and input and output variables the existence theorem of such interval, in which a continuous matrix of coefficients is identified, is formulated and proved. The parameter estimation algorithm, and also the question of the model validation are investigated on the example of the second order system identification.