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

By Igor Litvinchev, Vladimir Tsurkov (auth.)

When interpreting platforms with numerous parameters, the dimen­ sion of the unique approach could current insurmountable problems for the research. it will probably then be handy to reformulate the unique procedure by way of 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 comfortably 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 might 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 info and accuracy, getting better unique variables from aggregates, and so on. We think of additionally intimately the so-called iterative aggregation method. It constructs an iterative strategy, at· each step of which a macroproblem is solved that's less complicated than the unique challenge as a result of its decrease size. Aggregation weights are then up-to-date, and the process passes to the next move. Macrovariables are conventional in coordinating difficulties of hierarchical optimization.

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Extra info for [(Aggregation in Large-Scale Optimization )] [Author: Igor Litvinchev] [Sep-2003]

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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. Resolving the latter system with respect to V'j for dij = Pitj and dik = --1 . ( L bjdij )bk we obtam jETk It is not hard to verify that for this Vj we have L vjbj = Vkbk. Hence jETk v E D{v) and the claim follows.

This problem can be solved as follows. Let the weights be chosen as 9ij = bj/ :E bj, j E Tk. 8) CikXik, iES kETA L dikXik kETA LXik ~ ai, i E S, = bk (repeated ITkl times), k ETA, iES Xik ~ where Cik = :E JETk Cij9ij, dik = :E 0 Vi,k, dij9ij, bk JETk = :E JETk bj. 8) has an optimal solution. 9) iES kETA L dikXik kETA LXik ~ = bk, ai, i E S, k ETA, iES Xik ~ 0 Vi,k. 9) are the same, and the fixed-weight disaggregated solution Xij = 9 ij X i k, j E Tk is feasible to the §2. 1). 9). 1. 9}. Denote D(v) = {Vj, JET I L vjbj = Vkbk, k = 1, ...

5 max L: xj, k=l, ... ,K' jESk k = 1, ... , K' providing x* is known. 10) was rather moderate. 13) was enormous within the test examples. The change of the search direction from rays through the origin and the dual solution under consideration (Mendelssohn) to the direction given by the resource vector b (Shetty-Taylor) diminishes the bound significantly. The subgradient approach [14] to improve Zipkin bound provides the best results among dual-searching procedures. It has to be pointed out that the results of the numerical tests presented in [14, 17] have to be interpreted very carefully, and can not be treated as general results for the comparison of bounds behavior.

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