CORRESPONDENCE
T H E DISCRETE MAXIMUM PRINCIPLE SIR: We agree with the part of Horn and Jackson’s assertion (5, 6) that x l ~ v(the final value of the state variable to be maximized) has a stationary value with respect to variations in the 0’s if and only if each function H n (the Hamiltonian) is stationary with respect to variations in the corresponding Bn. However, the circumstances under which the .‘strong” result of Katz will be true can be much wider than the condition given by them that the Hessian of x I v reduces to the sum of the Hessians of Hn, n = 1, 2 , . . . -V. In fact, whenever the Hessians of x1” and Hn, as well as their every principal minor, ace of the same sign, the strong result of Katz will be valid. As far as we can see, all the solutions of the optimization problems published in this journal (8-70) and elsewhere ( 7 , 2) are correct for the following reasons. They belong to the classes of processes for which the strong result of Katz is valid. I n solving most of the problems, only the stationary condition of the Hamiltonian was used. Actually, the algorithm stated by Fan and Wang ( 4 ) is correct. although the proof requires some modifications, which we have given in our recent monograph (3). We would also like to call attention to a report by Jordan and Polak ( 7 ) . The following is directly quoted from the conclusions of their report. I t is interesting to see that Rozenoer’s assertion, that the “extension of the maximum principle to discrete systems is possible, generally speaking, only in the linear case,” is cor-
SIR: We agree, of course, with Fan and Wang’s assertion that the strong principle is true whenever “the Hessians of x l N and Hn, as well as their every principal minor, are of the same sign.” This is merely a restatement of the well known condition on second derivatives which determines the nature of a particular stationary value, and it forms the whole basis of our note (2). The more restrictive conditions we quote. on the other hand, identify classes of problems for which the strong principle is true, such as the linear and separate class quoted by Rozonoer (3). We regret that we are still unable to discern the truth of the algorithm stated in Equation 12 of Fan and Wang (4). The passage quoted from the report by Jordan and Polak confirms our contentions, and we would also agree with these writers’ comments on the value of the weak principle. It has not been our purpose to question the usefulness of the weak
rect and that the corresponding necessary conditions for the nonlinear case are, in fact, weaker than those given by Pontryagin-i.e., the Hamiltonian is required to be only a local maximum. In many systems, however, the Hamiltonian will have only one local maximum or stationary point and for these problems, the results here are as useful as those derived by Pon tryagin. Also noteworthy is the fact that Katz’s conclusion that the Hamiltonian must be a local maximum, is not quite complete due to his neglect of second-order terms. Rather, as shown in this paper, it is only necessary that the Hamiltonian be a local maximum or stationary. literature Cited
(1) Fan, L. T., Hwang, C. L., Wang, C. S., Eng. Expt. Sta., Kansas State University, Spec. Rept. 43 (1964). (2) Fan, L. T., Wang, C. S., Chem. Eng. Scz. 19, 86 (1964). (3) Fan, L. T., Wang, C. S., “Discrete Maximum Principle,” Wiley, New York, 1964. (4) Fan, L. T., Wang, C. S., J . Electron. Control 16, 441 (1964). (5) Horn, F., Jackson, R., IND.ENG.CHEM.FUNDAMENTALS 4, 110 (1965). (6) Horn, F., Jackson, R., J . Electron. Control, in press. (7) Jordan, B. W., Polak, E., “Theory of a Class of Discrete Optimal Control Systems,” Space Science Lab., University of California, Berkeley, Tech. Note, January 1964. (8) Katz, S., IND.ENG.CHEM.FUNDAMENTALS 1, 226 (1962). (9) Wang, C. S., Fan, L. T., Ibid.,3, 38 (1964). (10) Zahradnik, R. L., Archer, D. H., Ibzd., 2, 238 (1963).
L. T . Fan
Kansas State University Manhattan, Kan.
C. S. Wang
California Institute of Technology Pasadena, Calif.
principle [indeed it was derived and used by one of us ( 7 ) some time before any of the papers involved in the present correspondence were published], or of the applications published by Fan and Wang, but rather to increase the value of these methods by clarifying the extent to which they are valid.
literature Cited
(1) Horn, F., Chem. Eng. Sci. 15, 176 (1961). (2) Jackson, R., Horn, F., J . Electron. Contron, in press. (3) Rozenoer, L. I., Automation Remote Control 20, 1288, 1405, 1517 (1959).
F. Horn
Imperial College of Science and Technology London, England Vnzversity of Edinburgh and Edinburgh, Scotland
Heriot-W a t t College
VOL. 4
NO. 2
MAY
R. Jackson
1965
239
CORRESPONDENCE
DISCRETE MAXI MUM PRINCIPLE
SIR: Horn and Jackson (6) have made a valuable contribution in pointing out the distinction between optimization in discrete and continuous systems and the dangers inherent in the neglect of second-order terms, and we have been much indebted to them for this result in some of our own work on the theory of optimal processes (2-4). I t would appear that they have been unduly harsh, however, in their judgment of western workers as a consequence of their interpretation of Rozenoer’s statement that ... , . the extension of the maximum principle to discrete systems is possible, generally speaking, only in the linear case” ( 9 ) . Rozenoer’s remark has been understood to mean that, unlike the continuous case. the absolute maximum of the objective function may correspond only to a local maximum of the Hamiltonian. T h e Soviet, as well as the western literature, assumes incorrectly that the stationary points of the Hamiltonian must have the same character as the corresponding stationary points of the objective function, as is evident by the fact that so eminent a worker as Butkovskii has recently “proved” Rozenoer’s contention ( 7 ) and committed the identical error which was observed and corrected by Horn and Jackson. Also, Kushner and Schweppe (8) have recently “proved” a stochastic version of the discrete maximum principle which is incorrect as a consequence of failure to treat second-order terms. Similarly, Wang and Fan (70) have incorrectly ”proved” what is fortuitously a correct result for complex continuous systems ( 3 ) . If one adopts Horn’s approach to unconstrained discrete maximization problems (5, 7) and applies the Lagrange multiplier rule of differential calculus then the source of the difficulty becomes clear. Stationary points of the Hamiltonian are equivalent to stationary points of a Lagrangian function. But a proper formulation of the multiplier rule, apparently frequently misstated, is as follows: T h e unconstrained maxima (minima) of a function f ( x ) at constraints
240
I&EC FUNDAMENTALS
g‘(x) = 0; i = 1,2,.
. ., N
(1 1
will occur, when they exist, at stationary points of the Lagrangian N
L
=
c
f + r-1
hlgi
(2)
T h a t the stationary points need not be corresponding maxima or minima is easily verified by the example
f
= 41
+y)
g=x+yz
(3) (4)
I n the region -0.9 5 y 5 +O.l the function, f, has both a maximum and minimum at the constraint, g, while both stationary points of the Lagrangian are saddle points. literature Cited
(1) Butkovskii. A. G., Automation @ Remote Control 24, 963 (1964). (2) Denn, M . M., “The Optimization of Complex Systems,” Ph.D. thesis, University of Minnesota, Minneapolis, Minn., 1964. (3) Denn, M. M., Aris, Rutherford, Chem. Eng. Sci.,in press. (4) Denn. M. M., Aris, Rutherford, IND.ENG.CHEM.FUNDAMENTALS 4, 7 (1965). (5) Horn. F., Chem. Ens. Sci. 15, 176 (1961). 4, (6) Horn, F., Jackson, R., IND.ENG.CHEM.FUNDAMENTALS 110 11965). ~ NChem. , Eng. Sci. 19, 253 (1964). (7) J ~ C K S R., (8) Kushner, H. J., Schweppe, F. C., J . Math. Anal. Appl. 8, 287 (1964). (9) Rozenoer, L., Automation @ Remote Control 20, 1517 (1959). (10) Wang, C. S., Fan, L. T., J . Electron. Control 17, 199 (1964).
Morton M . Denn University of Delaware h k w a r k , Del.
