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

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

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**Sample text**

Stochastic Programming Models 53 For a discrete distribution of R we can convert the above mean–risk model into a linear programming problem. Indeed, let k ¼ 1, . . , K denote scenarios, and let Rik be the realization of the return P of security i in scenario k. The K probabilities of scenarios are p1 , . . , pK , k¼1 pk ¼ 1. Introducing new variables (representing the mean), and rk , k ¼ 1, . . 0, , r s:t: ð1 À Þ þ n X K X ) pk rk k¼1 i xi ¼ , i¼1 rk rk , k ¼ 1, . . , K, n X Rik xi , k ¼ 1, .

T À 1 the conditional distribution of t þ 1 given ½1, t ¼ ð1 , . . , t Þ is the same as the conditional distribution of t þ 1 given t . If the process 1 , . . , T is Markovian, the model is simpliﬁed considerably. 3) does not depend on 1 , . . 3) depends only on xTÀ2 and TÀ1 . Similarly, at stage t ¼ 2, . . 4) is then a function of xtÀ1 and t , and can be denoted by Qt ðxtÀ1 , t Þ. We shall call then t the information state of the model. In particular, the process 1 , . .

4) is equal to E½Qðx, Þ and the distribution of Qðx, Þ is symmetrical around its mean, then ðQðx, 1 Þ, . . , Qðx, K ÞÞ ¼ E½Qðx, Þ þ ð=2ÞVar½Qðx, Þ: Of course, the mean (expected value) of Qðx, Þ depends on x; in practical applications it would have to be iteratively adjusted during an optimization procedure. 3) is that then the function Ch. 1. 2) can be formulated as a linear programming problem. The above approach to stochastic programming is called robust by some authors. 4) is an example of a mean–risk model.