By A. Ruszczynski, and A. Shapiro (Eds.)

Brings jointly prime within the most crucial sub-fields of stochastic programming to give a rigourous review of easy versions, equipment and functions of stochastic programming. The textual content is meant for researchers, scholars, engineers and economists, who come upon of their paintings optimization difficulties related to uncertainty

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**Example text**

For instance, Pfb4 ¼ 10 j b3 ¼ 20, b2 ¼ 15, b1 ¼ 36g ¼ 0:5, while Pfb4 ¼ 10 j b3 ¼ 20g ¼ ¼ Pfb4 ¼ 10, b3 ¼ 20g Pfb3 ¼ 20g 0:5 Á 0:4 Á 0:4 þ 0:4 Á 0:4 Á 0:6 ¼ 0:44 6¼ 0:5: 0:4 Á 0:4 þ 0:4 Á 0:6 Ch. 1. Stochastic Programming Models 31 Fig. 3. Sequences of decisions for scenarios from Fig. 2. Horizontal dotted lines represent the equations of nonanticipativity. 10 For instance, E½b2 j b1 ¼ 36 ¼ E½b2 ¼ 15 Á 0:4 þ 50 Á 0:6 ¼ 36, E½b3 j b2 ¼ 15, b1 ¼ 36 ¼ 10 Á 0:1 þ 20 Á 0:4 þ 12 Á 0:5 ¼ 15, etc: Suppose now that cT ¼ 1 and AT , TÀ1 ¼ ATT ¼ 1.

T, in which st denotes the state of the system at time t, ut is the control vector, and et is a random ‘disturbance’ at time t. The matrices At , Bt and Ct are known. The random vectors et , t ¼ 1, . . , T, are assumed to be independent. At time t we observe the current state value, st , but not the disturbances et . Our objective is to ﬁnd a control law, u^ t ðÁÞ, t ¼ 1, . . , T, so that the actual values of the control variables can be determined through the feedback rule: ut ¼ u^ t ðst Þ, t ¼ 1, .

0, i ¼ 1, . . 21) becomes 38 A. Ruszczyn´ski and A. 21) becomes a deterministic optimization program. It has the trivial optimal solution of investing everything into the asset with the maximum expected return. Suppose, on the other hand, that UðWÞ is deﬁned as & UðWÞ :¼ ð1 þ qÞðW À aÞ, ð1 þ rÞðW À aÞ, if W ! a, if W a, ð3:22Þ with r > q > 0 and a > 0. We can view the involved parameters as follows: a is the amount that we have to pay at time t ¼ 1, q is the interest at which we can invest the additional wealth W À a, provided that W > a, and r is the interest at which we will have to borrow if W is less than a.