By Debora Mahlke

Optimization difficulties related to doubtful info come up in lots of components of business and financial purposes. Stochastic programming presents an invaluable framework for modeling and fixing optimization difficulties for which a chance distribution of the unknown parameters is available.

prompted through useful optimization difficulties taking place in power structures with regenerative power provide, Debora Mahlke formulates and analyzes multistage stochastic mixed-integer versions. for his or her answer, the writer proposes a singular decomposition procedure which depends on the idea that of splitting the underlying state of affairs tree into subtrees. in keeping with the formulated versions from power creation, the set of rules is computationally investigated and the numerical effects are discussed.

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**Extra resources for A Scenario Tree-Based Decomposition for Solving Multistage Stochastic Programs: With Application in Energy Production**

**Example text**

2. The path corresponding to the scenario s is highlighted in gray. Remember that a black node corresponds to an operating power plant and a white node to a plant which is switched oﬀ. 3. 2: Scenario tree representing the points bs,L and cs,L for L = 2 n n In order to verify the aﬃne independence of the 2N − 1 points, we subtract the point eoff from the other points show that they are linear independent. The resulting matrix is denoted by M := (M1 , M2 ), where M1 corresponds to the ﬁrst 2T − 2 columns of the matrix deﬁned by the up down s,L s,L points M1 = (aup , .

CkN −T , aup k1 , . . , akN −T ) leads to ⎛ C1 ⎜ ⎜ C2 ⎜ ⎜ ⎝ I N −T 0 A1 A2 0 I N −T ⎞ Ns ⎟ ⎫ ⎟ ⎪ ⎟ ⎪ ⎬ ⎟ N \ Ns ⎠ ⎪ ⎪ ⎭ Here, the ﬁrst line corresponds to all variables associated with scenario s. down with The lines two through four represent the variables xi , xup i and xi i ∈ N \Ns , respectively. Clearly, these points are linearly independent. down Observe that in matrix M1 all entries corresponding to xup varii or xi ables with i ∈ N \Ns equal zero. Hence, it follows that all columns of M are linearly independent and altogether we obtain 2N − 1 aﬃnely independent points.

K}. For each subinterval k ∈ P, we introduce a continuous variable δk and for each k ∈ P \ {K} a binary variable wk . Note that for the last subinterval K no binary variable wK is needed. In contrast to the textbook approach, a further binary variable z is included in order to ensure that f (x) equals zero if x = 0. 30b) 32 Chapter 3. Mathematical Modeling δk+1 ≤ 0 ≤ az ≤ δ1 ≤ wk ∈ z ∈ wk ≤ δk , δk ≤ 1, x ≤ bz, z, {0, 1}, {0, 1}. 30a) describes the variable x in dependence of δk . 30b). e.