Download e-book for kindle: Aggregation in Large-Scale Optimization by Igor Litvinchev, Vladimir Tsurkov (auth.)

By Igor Litvinchev, Vladimir Tsurkov (auth.)

ISBN-10: 1441991549

ISBN-13: 9781441991546

ISBN-10: 1461348129

ISBN-13: 9781461348122

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

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Example text

9) is equivalent to 0= Lx7 [afk(x)lESk aX l LUia~(X)]' aX i k=l, ... ,K. I Summing up these equalities, we get ' " ' " -k [ -k af (-) O =~~xl x aX l k lESk i ' " - aF (-)] =~Xj '" -~Uiakx i xl j [a f (_) x -a. X i xJ . The last equality holds since Uf[=l Sk = {I, ... , n} and clustering is simply a rearrangement of the indices of the original variables. 6) holds. Note that if f(x),Fi(X) are linear functions, then UB1(u, W) coincides with UB 2 (u, W). 1. Let some original constraints be included in the definition of W, such that W = {x I Fi(X)::; 0, i E I ~ {l, ...

Obviously, C-a(Wb n Wd) = C-a(Wb n Wd, 0), and we may try to decrease the upper bound C-a(Wb n Wd) by changing the duals starting from u = o. Suppose the bound c- a (Wb nWd) has been calculated and let be a unique optimal solution of the associated maximization problem. By the marginal value theorem [15] in this case C-a(Wb n Wd, u) is differentiable at u = 0, such that \7 uiC-a(Wb n Wd, 0) = -(L dijXij - ai). O C-a(Wb n Wd, u), we obtain x The stepsize can be defined, for example, by the one-dimensional search: p = argmin{c-a(WbnWd, u(p)) I p ~ a}.

Here laJ denotes the integer part of a. Thus there are a total of (3 problem sizes) x (5 cluster levels) x (10 problems)=150 observations for each class of the problems. Four localizations were used to calculate the aggregation bounds: Ws, Wd, Wb n Ws and Wb n Wd. For each localization the a priori bound c:a(W) and four a posteriori bounds c:i(W), c:~(W), ~T(W), and c:~(W) were calculated. These bounds were calculated for each problems class, except for the a priori bound, which was calculated only for the TP and SGTP.

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

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