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Hierarchical Optimization and Mathematical Physics (Paperback, Softcover reprint of the original 1st ed. 2000)
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Hierarchical Optimization and Mathematical Physics (Paperback, Softcover reprint of the original 1st ed. 2000)
Series: Applied Optimization, 37
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This book should be considered as an introduction to a special dass
of hierarchical systems of optimal control, where subsystems are
described by partial differential equations of various types.
Optimization is carried out by means of a two-level scheme, where
the center optimizes coordination for the upper level and
subsystems find the optimal solutions for independent local
problems. The main algorithm is a method of iterative aggregation.
The coordinator solves the problern with macrovariables, whose
number is less than the number of initial variables. This problern
is often very simple. On the lower level, we have the usual optimal
control problems of math ematical physics, which are far simpler
than the initial statements. Thus, the decomposition (or reduction
to problems ofless dimensions) is obtained. The algorithm
constructs a sequence of so-called disaggregated solutions that are
feasible for the main problern and converge to its optimal
solutionunder certain assumptions ( e.g., under strict convexity of
the input functions). Thus, we bridge the gap between two
disciplines: optimization theory of large-scale systems and
mathematical physics. The first motivation was a special model of
branch planning, where the final product obeys a preset assortment
relation. The ratio coefficient is maximized. Constraints are given
in the form of linear inequalities with block diagonal structure of
the part of a matrix that corresponds to subsystems. The central
coordinator assem bles the final production from the components
produced by the subsystems."
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