# Hierarchical Optimization and Mathematical Physics by Vladimir Tsurkov This publication can be regarded as an advent to a different dass of hierarchical structures of optimum keep watch over, the place subsystems are defined via partial differential equations of assorted forms. Optimization is performed via a two-level scheme, the place the guts optimizes coordination for the higher point and subsystems locate the optimum recommendations for autonomous neighborhood difficulties. the most set of rules is a technique of iterative aggregation. The coordinator solves the problern with macrovariables, whose quantity is below the variety of preliminary variables. This problern is usually extremely simple. at the decrease point, we now have the standard optimum regulate difficulties of math­ ematical physics, that are some distance easier than the preliminary statements. hence, the decomposition (or aid to difficulties ofless dimensions) is bought. The set of rules constructs a chain of so-called disaggregated options which are possible for the most problern and converge to its optimum solutionunder sure assumptions ( e.g., below strict convexity of the enter functions). therefore, we bridge the space among disciplines: optimization conception of large-scale structures and mathematical physics. the 1st motivation used to be a distinct version of department making plans, the place the ultimate product obeys a preset collection relation. The ratio coefficient is maximized. Constraints are given within the type of linear inequalities with block diagonal constitution of the a part of a matrix that corresponds to subsystems. The vital coordinator assem­ bles the ultimate construction from the parts produced through the subsystems.

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Extra resources for Hierarchical Optimization and Mathematical Physics

Example text

21) We enumerate sequentially all the elements of the set R and derrote the abtairred set of numbers by S = [1: S]. 13) and Chapter 1. The Main Model and Constructions 26 obtain the following expression: s ES. 22) Here, we assume a one-to-one correspondence between the elements of the sets R and S. 29) for fixed j E [1 : J]. Consider the following maximin problem: max. EMJ 7l'EI1 J"' h) = max. min 2::2::2:: :r:~EMJ 7l'EI1 . 7r 8 ~~(x;- ~;). 14, the problern (3. 2) has a saddle point, which we derrote by ( 1f8 ).

12), optimal solution 0 . {X~, 0 0. 26) was obtained, and xj is the 0, corresponding disaggregated solution. 26). 1) §3. 29) on the optimal and disaggregated solutions, respectively. We J L A set also h = 0 A hj, h = j=l J L 0 hj. 1. for all j E [1 : J]; " (b) h 0 = h "" 0 'if hj = hj for all jE [1: J]. 0· Proof. The disaggregated solution xj is feasible for local problems. 12). k ~i. k ci Xi = """ B~k Xi ~JJ ~JJ ~J iE Ij iE Ij < 1 _, i=I jE [1 : J], o. Since xj are feasible solutions of plant problems, and xj are their optimal solutions, the statement (a) of the lemma is proved.

T of the dual macroproblem, so that the monotonicity of the functional in the iterative process was not violated. In the second series, computation for the problern of dimension J = 20, I = 50, Kj = 23, j E [1 : 20] were carried out. The local problems were solved by simplex method. To find a saddle point of a maximin problem, we applied procedure based on the method of Brown. In the search of the 0 maximum of the function ()(pj) on the unit cube, it was assumed that pj = p, j E [1 : J]. For the one-dimensional extremum search, the golden section method was used.