By Ya. Z. Tsypkin
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1965). ” Nauka, Moscow. L. ( I 961). ” McGraw-Hill, New York. A. (1965). ” Academic Press, New York. R. (1959). “ T h e Theory of Matrices,” 2 Vols. Chelsea, New York. V. (1962). ” Chelsea, New York. S. (1961). ” Gosenergoizdat, Moscow. R. (1956). ” Tokyo. W. (1960). ” Pergamon Press, New York. Kalman, R. (1961). On the general theory of optimal control, “Proceedings of the First IFAC, Automatic and Remote Control,” Vol. 1, pp. 481-492. Butterworths, London. Kalman, R. (1964). When is a linear control system optimal?
12 Multistage Algorithms of Optimization 29 speleologist who explores a desert and tries to find its lowest point. He can examine or follow the local character only in the vicinity of his present position. How should he behave under these conditions? , the direction of the gradient). Then, he has to move in that direction as long as it continues to descend. The points of rest are actually the local minima. , narrow passages, exist, the tpeleologist has to follow them until he reaches the lowest point.
15) only are present. Using the method of Lagrange multipliers, we obtain the problem considered earlier. 24) where 3, = (Al, . . , A,) is still an unknown vector of Lagrange multipliers; T indicates the transport of a vector, and g(c) = ( g , ( c ) , . . ,g M ( c ) )is a vector function. 25) where is an N x A4 matrix. 25) by using the algorithms 2 Algorithmic Methods of Optimization 26 or 11 - r [ n ] V J ( ~ [-n 11 + G(c[~ - I]k[i? 28) The existence of equality constraints slightly complicates the structure of the system which corresponds to the algorithms of optimization.