By ICOLD Committee on Seismic Aspects of Dam Design
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Extra resources for ICOLD Position Paper On Dam Safety And Earthquakes
Sample text
The proof of Theorem 2 employs techniques used in Derman [15] and Gillette [33]. The proof of Corollary T+m 1 to Theorem 2 given here is a modification of the one given by Derman D81. For more on the remark following Theorem 2 see Derman [19] and Fisher and Ross [32]. Theorem 3 was proved by Derman [15]. In connection with Problem 2 following, Veinott [53] provides an alternative proof. His proof shows that the limit converges geometrically. This result in Problem 2 is used in the proof of Theorem 1 of Chapter 5.
In the limit, we get (2) with Ro E C, as that policy determined by those actions which minimize the right-hand side of (2). In practice, the limit is not attained, but a large number of iterations of (1) should in most cases yield the optimal policy or a good approximation. Policy Improvement Procedure We turn now to the policy improvement procedure for obtaining an optimal policy. Let R denote an arbitrary policy in C,. Then {aR(i))satisfies uniquely (Theorem 2 of Appendix A) the system For each i E I - (0) let Eidenote those actions a for which Define R E C, by choosing an action in E, for at least one i where Ei is not empty.
Maxlu,+,(i) - uL+l(i)/ S a maxlu,(i) - u,,'(i)l, is1 is1 Let ai',i c I , be the action which minimizes the right-hand side of Eq. (l), where in (1) u,,(i) is the nth iterate of uo'(i). Then Ia 1qij(ai') maxIu,(j) i i = a maxlu,(j) i - u,,'(j)l, - u,,'(j)I i E 1. Similarly, on letting a , , i E I , be the action which minimizes the righthand side of Eq. (l), where in ( 1 ) u,(i) is the nth iterate of uo(i), we obtain uA+ l(i) - on+ l ( i ) 5 a maxIu,(j) - u,,'(j)l, i i E I. Hence, we have shown the inequality.