By Igor Litvinchev, Vladimir Tsurkov (auth.)

ISBN-10: 1441991549

ISBN-13: 9781441991546

ISBN-10: 1461348129

ISBN-13: 9781461348122

When examining structures with loads of parameters, the dimen sion of the unique method might current insurmountable problems for the research. it could possibly then be handy to reformulate the unique procedure by way of considerably fewer aggregated variables, or macrovariables. In different phrases, an unique method with an n-dimensional vector of states is reformulated as a procedure with a vector of measurement less than n. The aggregated variables are both with ease outlined and processed, or the aggregated procedure can be regarded as an approximate version for the orig inal method. within the latter case, the operation of the unique approach should 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, recuperating unique variables from aggregates, and so forth. We think of additionally intimately the so-called iterative aggregation method. It constructs an iterative approach, at· each step of which a macroproblem is solved that's easier than the unique challenge due to its reduce measurement. Aggregation weights are then up-to-date, and the method passes to your next step. Macrovariables are common in coordinating difficulties of hierarchical optimization.

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Aggregated Problem and Bounds for Aggregation technique [23] and procedure is repeated to form the next set of variables, until no more covers can be found. The partition 0-' is then defined by the set of covers {Ql, Q2, ... } and the singleton sets of variables that are not included in a cover. A formal statement of the procedure can be found in [7]. We would like to stress the difference between the two considered approaches. In the first case a valid inequality is generated to cut off the disaggregated solution and the aim is to find the "strongest" inequality of this type.

Moreover, different bound computation techniques may result in different correspondence between the error bound and the actual error. We briefly discuss the results of a numerical study realized in [18] to test the bounds derived for the GTP and to explore the correspondence between the error bound and the actual error. Three classes of the problems were studied: the general GTP (GGTP), the specific GTP with dij = Pitj (SGTP), and the classical transportation problem (TP). For each class three problem sizes were considered, specified by (number of sources) x (number of destinations): small (30x50), medium (50xl00), and large (100xI50).

K, iES Xik ~ 0, where nk nk = = minjETk (1/ gj). If all gj are equal for each k = 1, ... 4). 1. Similar to Section 2, the choice of the weights in the form gj (independent on i) is aimed to achieve the redundancy in the column aggregated problem. This way a specific structure of the constraints of the GAP is preserved in the aggregated problem. 4). 1) can be used. Suppose that for each k = 1, ... , K the following subproblem was solved: Zk(X k ) = max L L CijXij, iES jETk L dijXij ~ dijX ik , i E S, jETk L Xij ~ 1, j E T k , iES XijE{O,I}, iES, jETk.

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

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