Many illdefined objects in macroeconomy, ecology, manufacturing
etc. can be described accurately enough by static algebraic or by
difference equations (with one or two delayed arguments only). This
peculiarity makes it possible to apply linear programming for normative
forecasting (after "whatif" scenario) and control optimization
of averaged variables.
Mathematical programming, linear or nonlinear, can be used for
optimization to find optimal values of input variables for each
set of output variables. The most difficult problem is to determine
necessary constrains of polygonal optimization area. Equations,
describing constrains and goal functions, can be obtained by GMDH
algorithm [1]. For application
in complex problems, the linear programming algorithm is to be revised
and modified in following directions [7,21]:
Instead of one scalar variable the vector of goal function
is to be used;
The procedure of tops polygonal area of constrains sortingout
is simplified.
Modifications in linear programming algorithm are following:
 Set of variables given in the data sampling, is divided
to two parts of input and output variables ( X = x_{1},x_{2},...,x_{M}
, Y = y_{1},y_{2},...,y_{L}). Subset
of output variables contains all of them, the reference optimal
values of which, are known. The problem is to find corresponding
optimal values of input variables;
 Each variable can be presented by its components. Therefore,
we always can simplify calculations by making M=L. Algorithm
is developed for this case only.
First step in algorithm is calculation of differences
R_{i} = y_{i}  y_{iopt} , i = 1,2,..,L
Second step is obtaining of models of constrains by Combinatorial
GMDH algorithm:
R_{i} = f_{i} (x_{1},x_{2},...,x_{M})
or
R_{i} = f_{i} (x_{1},x_{2},...,x_{M},
y_{1},y_{2},...,y_{i1},y_{i+1},...,y_{L}
)
Third step is system of constrains equations solution R_{i}
= 0. We obtain such way the solution of the problem
x_{1opt} x_{2opt} ... x_{3opt}
In the case, when object has inertia is necessary to include
as arguments into models of constrains the delayed values of
input variables. Instead of one instant we shall have twoinstant
models of constrains
R_{i} = f_{i} (x_{1(k)},...,x_{M(k)}
x_{1(k1)}...x_{M(k1)} x_{1(k2)}...x_{M(k2)})
Number of delayed argument, taking into account is to be slowly
increased, until the accuracy criterion RR decreases. Twoinstant
or threeinstant models of constrains are difference equations.
They can be used for stepbystep forecasting procedure. Such
forecast gives the answer to question, how the input variables
will change in time if the output variables will be kept on
optimal values (so called "normative forecasting").
Example of optimization of the world dynamics by multivariable
linear programming is given here.
The system of equations used in linear programming can be
considered as the first layer of neural network with active
neurons. If the error criterion is not small enough the next
layers of neuronet can be constructed to increase accuracy of
normative forecast and computer control. The use of optimal
nonphysical models for specification of constrains in linear
programing and selforganization of twicemultilayered
neuronet are two consistent steps in increase of accuracy
recommended for illdefined object control.
Examples of use of Simplified Linear Programming algorithm
should be used for computer advisers construction, normative
forecasting (after "whatifthen" scenario) and control optimization
of averaged variable.
