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  are the prescribed states of input nodes,
 is the vector characterizing the state of , ,  is the vector of , ,  is the
vector characterizing the state of controlling objects , . Thus, the state of the node depends on the state of
the nodes , and , , and is as well defined by the values of its parameters
.  is a function vector and generally it is nonlinear
with reference to its parameters.
Suppose that the Jacobian vector of function does not
degenerate at the assumed values of controlling actions,
consequently, states of , , nodes are not
defined uniquely under given , , and .
Denote via 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. .
Lemma 2. It follows from that and vice versa.
Lemma 3. It follows from that and consequently, .
Theorem 1. Vertex is incalculable recurrently,
if it is possible to find a loop cycle containing this vertex.
