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 non-linear
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.