Aggregation in Large-Scale Optimization by Igor Litvinchev, Vladimir Tsurkov (auth.)

By Igor Litvinchev, Vladimir Tsurkov (auth.)

When interpreting structures with various parameters, the dimen­ sion of the unique method might current insurmountable problems for the research. it may possibly then be handy to reformulate the unique process when it comes to considerably fewer aggregated variables, or macrovariables. In different phrases, an unique procedure with an n-dimensional vector of states is reformulated as a procedure with a vector of size less than n. The aggregated variables are both effectively outlined and processed, or the aggregated procedure will be regarded as an approximate version for the orig­ inal process. within the latter case, the operation of the unique process could be exhaustively analyzed in the framework of the aggregated version, and one faces the issues of defining the foundations for introducing macrovariables, specifying lack of details and accuracy, improving unique variables from aggregates, and so forth. We think of additionally intimately the so-called iterative aggregation strategy. It constructs an iterative approach, at· each step of which a macroproblem is solved that's easier than the unique challenge as a result of its reduce measurement. Aggregation weights are then up-to-date, and the process passes to the next move. Macrovariables are widespread in coordinating difficulties of hierarchical optimization.

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Aggregated Problem and Bounds for Aggregation The dimension of the original problem allows exact solution for all problem sizes. 1 the upper and the lower bounds for z* were characterized by (z-z*)jz* and (z* -LB)jz*. Here LB = z-c(W). 3 along with the corresponding standard deviations. 4 for the SGTP. 001 In all tables we use Wbs and Wbd to denote Wb n Ws and Wb n Wd, respectively. These preliminary computational results show that the error bounds calculated for the localization Wb n Wd are the tightest, compared with three other localizations.

Ax o ~ Xj ~ b, ~ 1 ===? L (xj)j ~ Pk, jES~ Xj integer, k = 1, ... , K'. 7). 1). It is really a clustering problem, since we wish to group together variables with approximately the same value of OJ. One possible heuristic procedure for performing such a clustering is to rank the variables according to OJ, and then to start from the largest value and include one after the other until the set of variables is a cover for one of the constraints. The cover is then reduced to a minimal by a standard 48 Chapter 1.

The Generalized Transportation Problem 31 gives an a priori bound for the classical TP. 14) does not allow to derive an a priori bound for the GTP. l. 1. 13) it is sufficient to choose the multipliers {Ui' Vj} in the form Ui = Ui, Vj : L vjbj = LVkbk. 10). 2. Let dij = Pitj, where Pi E D{v) such that for the pair u, we have v v > 0, tj > O. Then there exists Proof. 11) we have Ui 2: (Vk "Cik) /dik and hence -Uidij + V'j - Cij :::; [(Cikdij - Cijdik) - -{Vkdij - V'jdik)]/dik, j E T k · Now choose Vj to fulfill conditions Vkdij - Vjdik = 0 for all j E Tk.

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