On the Theorem of Bertini for Local Domains.

v1(°'(t), we obtain a system of linear equations whose solution is clearly xi = x,(°'. It follows that a set of forcing functions which minimize G subject to the linear equations of (2.1), together with the original constraints, yields a value of G which is at most G(x1(P)(T), . . . , xv(z)(T)). The general result follows inductively. This monotonicity is not surprising, since we are using the technique of approximation in policy space. 1 3. Successive Approximation and the Hitchcock-Koopmans Problem.-Let us now turn to the second problem described in section 1. As a first approximation, let XiJ be a set of values satisfying the constraints in (1.4). To obtain a second approximation, we fix the quantities sent out from the sources i = 3 to i = N, and determine the allocations from the first two sources so as to minimize the cost of supplying the remaining demand. This problem can be resolved in terms of sequences of functions of one variable.6 To obtain a third approximation, we fix the allocations from the first source and the sources i = 4 to i = N, and determine the allocation from the second and third sources so as to minimize the cost of supplying the remaining demands. Continuing in this fashion, we obtain a sequence of problems, each of whose solutions depends upon a sequence of functions of one variable. As above, it is easy to see that the sequence of costs obtained in this way is monotone decreasing. Once again, interesting questions arise concerning convergence which we do not enter into here. Similar techniques can be applied to other classes of combinatorial problems as will be shown elsewhere. 'R. Bellman, Dynamic Programming (Princeton: Princeton University Press, 1957). 2 R. Bellman, "Terminal Control, Time Lags, and Dynamic Programming," these PROCEEDINGS, 43,927-930, 1957. 3 R. Bellman, "Some New Techniques in the Dynamic Programming Solution of Variational Problems," Quart. Appi. Math. (to appear). 4 R. Bellman, I. Glicksberg, and 0. Gross, "On Some Variational Problems Occurring in the Theory of Dynamic Programming," Rend. Palermo, Ser. II, 3, 1-35, 1954. 5 R. Bellman, W. H. Fleming, and D. V. Widder, "Variational Problems with Constraints Ann. di Mat., Ser. IV, 49, 301-323,1956. 6 R. Bellman, "Notes on the Theory of Dynamic Programming-Transportation Models." Management Sci., 4, 191-195, 1958.