19.2. MAT H E M AT IC A L PROGRAMMING 1025 where a ij is the payoff (positive or negative) of player A against player B if player A uses the pure strategy A i and player B used the pure strategy B j . Remark. The sum of payoffs of both players is zero for each move. (That is why the game is called a zero-sum game.) Let α i =min j {a ij } be the minimum possible payoff of player A if he uses the pure strategy A i .IfplayerA acts reasonably, he must choose a strategy A i for which α i is maximal, α =max i {α i } =max i min j {a ij }.(19.2.1.28) The number α is called the lower price of the game.Letβ j =max i a ij be the maximum possible loss of player B if he uses the pure strategy B j .IfplayerB acts reasonably, he must choose a strategy B j for which β j is minimal, β =min j β j =min j max i {a ij }.(19.2.1.29) The number β is called the upper price of the game. Remark. The principle for constructing the strategies of player A (the first player) based on the maxi- mization of minimal payoffs is called the maximin principle. The principle for constructing the strategies of player B (the second player) based on the minimization of maximal losses is called the minimax principle. The lower price of the game is the guaranteed minimal payoff of player A if he follows the maximin principle. The upper price of the game is the guaranteed maximal loss of player B if he follows the minimax principle. T HEOREM. In a two-person zero-sum game, the lower price α and the upper price β satisfy the inequality α ≤ β.(19.2.1.30) If α =β, then such a game is called the game with a saddle point,andapair(A i,opt , B j,opt ) of optimal strategies is called a saddle point of the payoff matrix.Theentryv = a ij corresponding to a saddle point (A i,opt , B j,opt ) is called the game value. If a game has a saddle point, then one says that the game can be solved in pure strategies. Remark. There can be several saddle points, but they all have the same value. If the payoff matrix has no saddle points, i.e., the strict inequality α < β holds, then the search of a solution of the game leads to a complex strategy in which a player randomly uses two or more strategies with certain frequencies. Such complex strategies are said to be mixed. The strategies of player A are determined by the set x =(x 1 , , x m ) of probabilities that the player uses the respective pure strategies A 1 , , A m . For player B, the strategies are determined by the set y =(y 1 , , y n ) of probabilities that the player uses the respective pure strategies B 1 , , B n . These sets of probabilities must satisfy the identity m i=1 x i = n j=1 y j = 1. The expectation of the payoff of player A is given by the function H(x, y)= m i=1 n j=1 a ij x i y j .(19.2.1.31) 1026 CALCULUS OF VARIATIONS AND OPTIMIZATION THE VON NEUMANN MINIMAX THEOREM. There exist optimal mixed strategies x ∗ and y ∗ , i.e., strategies such that H(x, y ∗ ) ≤ H(x ∗ , y ∗ ) ≤ H(x ∗ , y)(19.2.1.32) for any probabilities x and y . The number v = H(x ∗ , y ∗ ) is called the game price in mixed strategies. M INIMAX THEOREM FOR ANTAGONISTIC TWO-PERSON ZERO-SUM GAMES. For any payoff matrix (19.2.1.27), v =max x 1 , ,x m min y 1 , ,y n m i=1 n j=1 a ij x i y j =min y 1 , ,y n max x 1 , ,x m m i=1 n j=1 a ij x i y j .(19.2.1.33) 19.2.1-8. Relationship between game theory and linear programming. Without loss of generality, we can assume that v > 0. This can be ensured if we add the same positive constant a > 0 to all entries a ij of the payoff matrix (19.2.1.27); in this case, only the game price varies (increases by a > 0), while the optimal solution remains the same. An antagonistic two-person zero-sum game can be reduced to a linear programming problem by the change of variables v = 1 Z min = 1 W max , x i = vX i (i = 1, 2, , m); y j = vY j (j = 1, 2, , n). (19.2.1.34) The quantities Z min , W max , X i ,andY j form a solution of the following pair of dual problems: Z = X 1 + X 2 + ···+ X m → min, a 11 X 1 + a 21 X 2 + ···+ a m1 X m ≥ 1, a 12 X 1 + a 22 X 2 + ···+ a m2 X m ≥ 1, a 1n X 1 + a 2n X 2 + ···+ a mn X m ≥ 1, X i ≥ 0 (i = 1, 2, , m); (19.2.1.35) W = Y 1 + Y 2 + ···+ Y n → max, a 11 Y 1 + a 12 Y 2 + ···+ a 1n Y n ≤ 1, a 21 Y 1 + a 22 Y 2 + ···+ a 2n Y n ≤ 1, a m1 Y 1 + a m2 Y 2 + ···+ a mn Y n ≤ 1, Y j ≥ 0 (j = 1, 2, , n). (19.2.1.36) 19.2. MAT H E M AT IC A L PROGRAMMING 1027 19.2.2. Nonlinear Programming 19.2.2-1. General statement of nonlinear programming problem. The nonlinear programming problem is the problem of finding n variables x =(x 1 , , x n ) that provide an extremum of the objective function Z(x)=f(x) → extremum (19.2.2.1) and satisfy the system of constraints ϕ i (x)=0 for i = 1, 2, , k, ϕ i (x) ≤ 0 for i = k + 1, k + 2, ,l, ϕ i (x) ≥ 0 for i = l + 1, l + 2, , m. (19.2.2.2) Here the objective function (19.2.2.1) and/or at least one of the functions ϕ i (x)(i = 1, 2, , m) is nonlinear. Depending on the properties of the functions f(x)andϕ i (x), the following types of problems are distinguished: 1. Convex programming. 2. Quadratic programming. 3. Geometric programming. A necessary condition for the maximum of the function Z(x)=f (x)(19.2.2.3) under the inequality constraints ϕ i (x) ≤ 0 (i = 1, 2, , m) is that there exist m + 1 nonnegative Lagrange multipliers λ 0 , λ 1 , , λ m that are not simultaneously zero and satisfy the conditions λ i ≥ 0 (i = 0, 1, , m), λ i ϕ i (x)=0 (i = 1, 2, , m), λ 0 f x j + m i=1 λ i (ϕ i ) x j = 0, (19.2.2.4) where derivatives f x j and (ϕ i ) x j are evaluated at x. One of the most widely used methods of nonlinear programming is the penalty function method. This method approximates a problem with constraints by a problem without constraints and with objective function that penalizes infeasibility. The higher the penalties, the closer the problem of maximizing the penalty function is to the original problem. 19.2.2-2. Dynamic programming. Dynamic programming is the branch of mathematical programming dealing with multistage optimal decision-making problems. The general outline of a multistage optimal decision-making process is as follows. Consider a controlled system S taken by the control from an initial state s 0 to a state s.Let 1028 CALCULUS OF VARIATIONS AND OPTIMIZATION x k (k = 1, 2, , n) be the control at the kth stage, let x =(x 1 , , x n ) be the control taking the system S from the state s 0 to the state s,andlets k be the state of the system after the kth control step. The efficiency of the control is characterized by an objective function that depends on the initial state and the control, Z = F (s 0 , x). (19.2.2.5) We assume that 1. The state s k depends only on the preceding state s k–1 and the control x k at the kth step, s k = ϕ k (s k–1 , x k )(k = 1, 2, , n). (19.2.2.6) 2. The objective function (19.2.2.5) is an additive function of the performance factor at each step. If the performance factor at the kth step is Z k = f k (s k–1 , x k )(k = 1, 2, , n), (19.2.2.7) then the objective function (19.2.2.5) can be written as Z = n k=1 f k (s k–1 , x k ). (19.2.2.8) The dynamic programming problem. Find an admissible control x taking the sys- tem S from the state s 0 to the state s and maximizing (or minimizing) the objective function (19.2.2.8). T HEOREM (BELLMAN’S OPTIMALITY PRINCIPLE). For any state s of the system after any number of steps, one should choose the control at the current step so as to ensure that this control, together with the optimal control at all subsequent steps, leads to the optimal payoff at all remaining steps, including the current step. Let Z ∗ k (s k–1 ) be the conditional maximum of the objective function obtained under the optimal control at n–k–1 steps starting from the kth step until the end under the assumption that the system was in the state s k–1 by the beginning of the kth step. The equations Z ∗ n (s n–1 )=max x n {f n (s n–1 , x n )}, Z ∗ k (s k–1 )=max x k {f k (s k–1 , x k )+Z ∗ k+1 (s k )} (k = n – 1, n – 2, , 1) are called the Bellman equations. The Bellman equations for the dynamic programming problem and for any n and s 0 permit finding a solution, which is given by Z max = Z ∗ 1 (s 0 ). References for Chapter 19 Akhiezer, N. I., Calculus of Variations, Taylor & Francis, London, New York, 1988. Avriel, M., Nonlinear Programming: Analysis and Methods, Dover Publications, New York, 2003. Bazaraa, M. S., Sherali, H. D., and Shetty, C. M., Nonlinear Programming: Theory and Algorithms, Wiley, New York, 2006. Belegundu, A. D. and Chandrupatla, T. R., Optimization Concepts and Applications in Engineering, Bk&CD ROM Edition, Prentice Hall, Englewood Cliffs, New Jersey, 1999. REFERENCES FOR CHAPTER 19 1029 Bellman, R., Adaptive Control Processes: A Guided Tour, Princeton University Press, Princeton, New Jersey, 1961. Bellman, R., Dynamic Programming, Dover Edition, Dover Publications, New York, 2003. Bellman, R. and Dreyfus, S. E., Applied Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1962. Bertsekas, D. P., Nonlinear Programming, 2nd Edition, Athena Scientific, Belmont, Massachusetts, 1999. Bertsimas, D. and Tsitsiklis, J. N., Introduction to Linear Optimization (Athena Scientific Series in Optimiza- tion and Neural Computation, Vol. 6), Athena Scientific, Belmont, Massachusetts, 1997. Bolza, O., Lectures on the Calculus of Variations, 3rd Edition, American Mathematical Society, Providence, Rhode Island, 2000. Bonnans, J. F., Gilbert, J. C., Lemarechal, C., and Sagastizabal, C. A., Numerical Optimization, Springer, New York, 2003. Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press, Cambridge, 2004. Brechteken-Mandersch, U., Introduction to the Calculus of Variations, Chapman & Hall/CRC Press, Boca Raton, 1991. Brinkhuis, J. and Tikhomirov, V., Optimization: Insights and Applications, Princeton University Press, Princeton, New Jersey, 2005. Bronshtein, I. N., Semendyayev, K. A., Musiol, G., and M ¨ uhlig, H., Handbook of Mathematics, 4th Edition, Springer, New York, 2004. Bronson, R. and Naadimuthu, G., Schaum’s Outline of Operations Research, 2nd Edition, McGraw-Hill, New York, 1997. van Brunt, B., The Calculus of Variations, Springer, New York, 2003. Calvert, J. E., Linear Programming, Brooks Cole, Stamford, 1989. Chong, E. K. P. and ˇ Zak, S. H., An Introduction to Optimization, 2nd Edition, Wiley, New York, 2001. Chvatal, V., Linear Programming (Series of Books in the Mathematical Sciences), W. H. Freeman, New York, 1983. Cooper, L., Applied Nonlinear Programming for Engineers and Scientists, Aloray, Goshen, 1974. Dacorogna, B., Introduction to the Calculus of Variations, Imperial College Press, London, 2004. Darst, R. B., Introduction to Linear Programming (Pure and Applied Mathematics (Marcel Dekker)), CRC Press, Boca Raton, 1990. Denardo, E. V., Dynamic Programming: Models and Applications, Dover Publications, New York, 2003. Dreyfus, S. E., Dynamic Programming and the Calculus of Variations, Academic Press, New York, 1965. Elsgolts, L., Differential Equations and the Calculus of Variations, University Press of the Pacific, Honolulu, Hawaii, 2003. Ewing, G. M., Calculus of Variations with Applications (Mathematics Series), Dover Publications, New York, 1985. Fletcher, R., Practical Methods of Optimization, 2nd Edition, Wiley, New York, 2000. Fomin, S. V. and Gelfand, I. M., Calculus of Variations, Dover Publications, New York, 2000. Fox, C., An Introduction to the Calculus of Variations, Dover Publications, New York, 1987. Galeev, E. M. and Tikhomirov, V. M., Optimization: Theory, Examples, and Problems [in Russian], Editorial URSS, Moscow, 2000. Gass, S. I., An Illustrated Guide to Linear Programming, Rep. Edition, Dover Publications, New York, 1990. Gass, S. I., Linear Programming: Methods and Applications, 5th Edition, Dover Publications, New York, 2003. Gill, Ph. E., Murray, W., and Wright, M. H., Practical Optimization, Rep. Edition, Academic Press, New York, 1982. Giusti, E., Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Hackensack, New Jersey, 2003. Glicksman, A. M., Introduction to Linear Programming and the Theory of Games, Dover Publications, New York, 2001. Hillier, F. S. and Lieberman, G. J., Introduction to Operations Research, McGraw-Hill, New York, 2002. Horst, R. and Pardalos, P. M. (Editors), Handbook of Global Optimization, Kluwer Academic, Dordrecht, 1995. Jensen,P.A.andBard,J.F.,Operations Research Models and Methods, Bk&CD-Rom Edition, Wiley, New York, 2002. Jost, J. and Li-Jost, X., Calculus of Variations (Cambridge Studies in Advanced Mathematics), Cambridge University Press, Cambridge, 1999. Kolman,B.andBeck,R.E.,Elementary Linear Programming with Applications, 2nd Edition (Computer Science and Scientific Computing), Academic Press, New York, 1995. Korn,G.A.andKorn,T.M.,Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 2nd Rev. Edition, Dover Publications, New York, 2000. 1030 CALCULUS OF VARIATIONS AND OPTIMIZATION Krasnov, M. L., G. K., Makarenko, G. K.,, and Kiselev, A. I., Problems and Exercises in the Calculus of Variations, Imported Publications, Inc., New York, 1985. Lebedev, L. P. and Cloud, M. J., The Calculus of Variations and Functional Analysis with Optimal Control and Applications in Mechanics (Series on Stability, Vibration and Control of Systems, Series A, Vol. 12), World Scientific Publishing Co., Hackensack, New Jersey, 2003. Liberti, L. and Maculan, N. (Editors), Global Optimization: From Theory to Implementation (Nonconvex Optimization and Its Applications), Springer, New York, 2006. Luenberger, D. G., Linear and Nonlinear Programming, 2nd Edition, Springer, New York, 2003. MacCluer, C. R., Calculus of Variations: Mechanics, Control, and Other Applications, Prentice Hall, Engle- wood Cliffs, New Jersey, 2004. Mangasarian,O. L., Nonlinear Programming (Classics in Applied Mathematics, Vol. 10),Society for Industrial & Applied Mathematics, University City Science Center, Philadelphia, 1994. Marlow, W. H., Mathematics for Operations Research, Dover Publications, New York, 1993. Morse, Ph. M. and Kimball, G. E., Methods of Operations Research, Dover Publications, New York, 2003. Moser, J., Selected Chapters in the Calculus of Variations: Lecture Notes by Oliver Knill (Lectures in Mathe- matics. ETH Zurich), Birkh ¨ auser Verlag, Basel, Stuttgart, 2003. Murty, K. G., Linear Programming, Rev. Edition, Wiley, New York, 1983. Nash,S.G.andSofer,A.,Linear and Nonlinear Programming, McGraw-Hill, New York, 1995. Nocedal, J. and Wright, S., Numerical Optimization, Springer, New York, 2000. Padberg, M., Linear Optimization and Extensions (Algorithms and Combinatorics), 2nd Edition, Springer, New York, 1999. Pannell, D. J., Introduction to Practical Linear Programming, Bk&Disk Edition, Wiley, New York, 1996. Pardalos,P.M.andResende,M.G.C.(Editors),Handbook of Applied Optimization, Oxford University Press, Oxford, 2002. Pardalos, P. M. and Romeijn, H. E. (Editors), Handbook of Global Optimization, Vol. 2 (Nonconvex Opti- mization and Its Applications), Springer, New York, 2002. Pierre, D. A., Optimization Theory with Applications, Dover Publications, New York, 1987. Rao, S. S., Engineering Optimization: Theory and Practice, 3rd Edition, Wiley, New York, 1996. Rardin,R.L.,Optimization in Operations Research, Prentice Hall, Englewood Cliffs, New Jersey, 1997. Ross, S. M., Applied Probability Models with Optimization Applications, Rep. Edition (Dover Books on Mathematics), Dover Publications, New York, 1992. Ruszczynski, A., Nonlinear Optimization, Princeton University Press, Princeton, New Jersey, 2006. Sagan, H., Introduction to the Calculus of Variations, Rep. Edition, Dover Publications, New York, 1992. Shenoy,G.V.,Linear Programming: Methods and Applications, Halsted Press, New York, 1989. Simon, C. P. and Blume, L., Mathematics for Economists, W. W. Norton & Company, New York, 1994. Smith, D. R., Variational Methods in Optimization (Dover Books on Mathematics), Dover Publications, New York, 1998. Strayer, J. K., Linear Programming and Its Applications (Undergraduate Texts in Mathematics), Springer- Verlag, Berlin, 1989. Sundaram, R. K., A First Course in Optimization Theory, Cambridge University Press, Cambridge, 1996. Taha, H. A., Operations Research: An Introduction, 7th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2002. Tslaf, L. Ya., Calculus of Variations and Integral Equations, 3rd Edition [in Russian], Lan, Moscow, 2005. Tuckey,C., NonstandardMethods intheCalculus of Variations, Chapman& Hall/CRC Press, BocaRaton,1993. Vasilyev,F.P.andIvanitskiy,A.Y.,In-Depth Analysis of Linear Programming, Springer, New York, 2001. Venkataraman, P., Applied Optimization with MATLAB Programming, Wiley, New York, 2001. Wan, F., Introduction to the Calculus of Variations and Its Applications, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 1995. Weinstock, R., Calculus of Variations, Dover Publications, New York, 1974. Winston, W. L., Operations Research: Applications and Algorithms (with CD-ROM and InfoTrac), 4th Edition, Duxbury Press, Boston, 2003. Young,L.C.,Lecture on the Calculus of Variations and Optimal Control Theory, American Mathematical Society, Providence, Rhode Island, 2000. Chapter 20 Probability Theory 20.1. Simplest Probabilistic Models 20.1.1. Probabilities of Random Events 20.1.1-1. Random events. Basic definitions. The simplest indivisible mutually exclusive outcomes of anexperiment arecalled elementary events ω. The set of all elementary outcomes, which we denote by the symbol Ω, is called the space of elementary events or the sample space. Any subset of Ω is called a random event A (or simply an event A). Elementary events that belong to A are said to favor A.In any probabilistic model, a certain condition set Σ is assumed to be fixed. An event A implies an event B (A ⊆B)ifB occurs in each realization of Σ for which A occurs. Events A and B are said to be equivalent (A = B)ifA implies B and B implies A, i.e., if, for each realization of Σ, both events A and B occur or do not occur simultaneously. The intersection C = A ∩ B = AB of events A and B is the event that both A and B occur. The elementary outcomes of the intersection AB are the elementary outcomes that simultaneously belong to A and B. The union C = A ∪ B = A + B of events A and B is the event that at least one of the events A or B occurs. The elementary outcomes of the union A + B are the elementary outcomes that belong to at least one of the events A and B. The difference C = A\B = A–B of events A and B is the event that A occurs and B does not occur. The elementary outcomes of the difference A\B are the elementary outcomes of A that do not belong to B. The event that A does not occur is called the complement of A,orthecomplementary event, and is denoted by A. The elementary outcomes of A are the elementary outcomes that do not belong to the event A. An event is said to be sure if it necessarily occurs for each realization of the condition set Σ. Obviously, the sure event is equivalent to the space of elementary events, and hence the sure event should be denoted by the symbol Ω. An event is said to be impossible if it cannot occur for any realization of the condition set Σ. Obviously, the impossible event does not contain any elementary outcome and hence should be denoted by the symbol ∅. Two events A and A are said to be opposite if they simultaneously satisfy the following two conditions: A ∪ A = Ω, A ∩A = ∅. Events A and B are said to be incompatible,ormutually exclusive, if their simultaneous realization is impossible, i.e., if A ∩B = ∅. Events H 1 , , H n are said to form a complete group of events,ortobecollectively exhaustive, if at least one of them necessarily occurs for each realization of the condition set Σ, i.e., if H 1 ∪···∪H n = Ω. 1031 . 1996. Pardalos,P.M.andResende,M.G.C.(Editors) ,Handbook of Applied Optimization, Oxford University Press, Oxford, 2002. Pardalos, P. M. and Romeijn, H. E. (Editors), Handbook of Global Optimization, Vol. 2 (Nonconvex Opti- mization and. Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 2nd Rev. Edition, Dover Publications, New York, 2000. 1030 CALCULUS OF VARIATIONS AND OPTIMIZATION Krasnov,. payoff (positive or negative) of player A against player B if player A uses the pure strategy A i and player B used the pure strategy B j . Remark. The sum of payoffs of both players is zero for