2 The constraint matrix W t is block diagonal W t A t B t 5 5 3 The other

2 the constraint matrix w t is block diagonal w t a t

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2. The constraint matrix W t is block diagonal: W t = A t 0 0 B t . (5 . 5) 3. The other components of the constraints are random but we assume that for each realization of ω , T t ( ω ) and h t ( ω ) can be written: T t ( ω ) = R t ( ω ) 0 S t ( ω ) 0 and h t ( ω ) = b t ( ω ) d t ( ω ) , (5 . 6) where the zero components of T t correspond to the detailed level variables. Notice that (3) in the definition implies that detailed level variables have no direct effect on future constraints. This is the fundamental advantage of block separability. With block separable recourse, we may rewrite Q t ( x t 1 , ξ t ( ω )) as the sum of two quantities, Q t w ( w t 1 , ξ t ( ω )) + Q t y ( w t 1 , ξ t ( ω )), where we need
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132 3. Basic Properties and Theory not include the y t 1 terms in x t 1 , Q t w ( w t 1 , ξ t ( ω )) = min r t ( ω ) w t ( ω ) + Q t +1 ( x t ) s . t . A t w t ( ω )= b t ( ω ) R t 1 ( ω ) w t 1 , w t ( ω ) 0 , (5 . 7) and Q t y ( w t 1 , ξ t ( ω )) = min q t ( ω ) y t ( ω ) s . t . B t y t ( ω )= d t ( ω ) S t 1 ( ω ) w t 1 , y t ( ω ) 0 . (5 . 8) The great advantage of block separability is that we need not consider nesting among the detailed level decisions. In this way, the w variables can all be pulled together into a first stage of aggregate level decisions. The second stage is then composed of the detailed level decisions. Note that if the b t and R t are known, then the block separable problem is equivalent to a similarly sized two-stage stochastic linear program. Separability is indeed a very useful property for stochastic programs. Computational methods should try to exploit it whenever it is inherent in the problem because it may reduce work by orders of magnitude. We will also see in Chapter 11 that separability can be added to a problem (with some error that can be bounded). This approach opens many possible applications with large numbers of random variables. Another modeling approach that may have some computational advan- tage appears in Grinold [1976]. This approach extends from analyses of stochastic programs as examples of Markov decision process. He assumes that ω t belongs to some finite set 1 , . . . , k t , that the probabilities are deter- mined by p ij = P { ω t +1 = j | ω t = i } for all t , and that T t = T t ( ω t , ω t +1 ). In this framework, he can obtain an approximation that again obtains a form of separability of future decisions from previous outcomes. We discuss more approximation approaches in Chapter 11. We now consider generalizations into nonlinear functions and infinite horizons. The general results of the previous section can be extended here directly. We concentrate on some areas where differences may oc- cur and, for notational convenience, concentrate just on the description of problems in the form with explicit nonanticipativity constraints. More detailed descriptions of these problems appear in the papers by Rockafel- lar and Wets [1976a,1976b], Dempster [1988], Fl˚ am [1985], and Birge and Dempster [1992].
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