20th GAMM-Seminar Leipzig on
Numerical Methods for Non-Local Operators

Max-Planck-Institute for Mathematics in the Sciences
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  20th GAMM-Seminar
January, 22th-24th, 2004
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  The Method of Solution of Optimization Network Mathematical Model
Seilkhan Boranbayev (University Kazakhstan)

The mathematical model of M graph can be presented as follows:


where tex2html_wrap_inline22 - are the prescribed states of input nodes, tex2html_wrap_inline24 - is the vector characterizing the state of tex2html_wrap_inline26, tex2html_wrap_inline28, tex2html_wrap_inline30 - is the vector of tex2html_wrap_inline26, tex2html_wrap_inline28, tex2html_wrap_inline36 - is the vector characterizing the state of controlling objects tex2html_wrap_inline38, tex2html_wrap_inline40. Thus, the state of the tex2html_wrap_inline26 node depends on the state of the nodes tex2html_wrap_inline44, tex2html_wrap_inline46 and tex2html_wrap_inline38, tex2html_wrap_inline50, and is as well defined by the values of its parameters tex2html_wrap_inline30. tex2html_wrap_inline54 - is a function vector and generally it is non-linear with reference to its parameters.

Suppose that the Jacobian vector of tex2html_wrap_inline54 function does not degenerate at the assumed values of controlling actions, consequently, tex2html_wrap_inline24 states of tex2html_wrap_inline26, tex2html_wrap_inline28, nodes are not defined uniquely under given tex2html_wrap_inline22, tex2html_wrap_inline66, and tex2html_wrap_inline68.

Denote via tex2html_wrap_inline70 the set of vertices the conditions of which are defined by recurrent relations (i.e. these are the vertices which do not require solution of some subsystem equations (1) - (2) for the definition of their states).

Lemma 1. The set of a graph's inputs is defined by recurrent relations, i.e. tex2html_wrap_inline72.

Lemma 2. It follows from tex2html_wrap_inline74 that tex2html_wrap_inline76 and vice versa.

Lemma 3. It follows from tex2html_wrap_inline78 that tex2html_wrap_inline76 and consequently, tex2html_wrap_inline74.

Theorem 1. Vertex tex2html_wrap_inline84 is incalculable recurrently, if it is possible to find a loop cycle containing this vertex.

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