Download SOME DISCRETE EXTREME PROBLEMS

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SOME DISCRETE EXTREME PROBLEMS
IN THE RECOGNITION THEORY1
N.N. Katerinochkina2
2Dorodnicyn
Computing Centre, Russian Academy of Science,
ul. Vavilova 40, GSP-1, Moscow, 119991 Russia,
e-mail: [email protected]
By optimization of different models of recognition algorithms a number of discrete
extreme problems appears. The search for the maximum solvable subsystem of the
system of linear inequalities is one of such tasks. The approximate method for solution
of this problem, effective for the systems of large dimensionality is proposed.
Introduction
Solution of optimization problems in the
recognition theory is one of the most important
stages of synthesis of the high-precision
algorithms of recognition and forecast. During
optimization of some models of recognition
algorithms (for example, the estimatecalculating algorithms - ECA) it is required as
it is possible to more precisely satisfy the
specific system of conditions, which is
described by the large number of linear
inequalities and as a whole it can be
contradictory. In the general case the solution
of this problem is reduced to the search for
maximal (according to the number of
inequalities) solvable subsystem of the
assigned system of linear inequalities.
For some models of algorithms additional
requirements on a desired solvable subsystem
are imposed. For example, for ECA such
system can reflect a condition for the correct
distribution of objects of control sample into
classes according to decision rule. In this case
the given system of linear inequalities is
broken into the blocks (subsystems) of
identical power. The number of such blocks
coincides with the number of objects of
control sample and each block determines the
condition for correct recognition for one
control object. Thus, j-th object will be
correctly recognized, if all inequalities of j-th
block are fulfilled. The solution of this
optimization problem means a finding of the
solvable subsystem, which corresponds to the
maximum number of correctly recognized
objects. Consequently, it is necessary to find
the solvable subsystem, which contains the
maximum number of complete blocks.
In other cases it is necessary to find the
maximum solvable subsystem, which contains
the fixed inequalities of the assigned system,
and etc.
The search methods for the maximum
solvable subsystem of the system of linear
inequalities
We consider the general task of the search for
the maximum solvable subsystem (MSS) of
the system of linear inequalities. Let us assume
that the inconsistent system of the linear
inequalities of the following form is assigned:
n
a x
j 1
j
 bi , i  1,..., m.
(1)
It is necessary to find MSS of this system.
The approach to the solution of this problem,
using method of convoluting of the system of
linear inequalities is known (see [1]).
However, this method requires the rapidly
growing storage capacities and is unfit for the
tasks of large dimensionality.
_______________________________________________________________________
1
ij
This work was supported by the RFBR (projects 05-01-00332, 05-07-90333).
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The algorithm of the solution of this problem,
effective for linear systems of a small rank
with large number of inequalities had been
previously constructed (see [2]). This
algorithm is based on following assertions.
Suppose that the unsolvable system S of form
(1) is given.
DEFINITION 1. We will call the solvable
subsystem of system S not extensible solvable
subsystem, if the addition to it of any (not
entering it) inequality of system S makes with
its inconsistent.
The following assertion is true.
LEMMA 1. Assume that the inconsistent
system S has a rank r, r > 0. Then rank of any
not extensible solvable subsystem of system S
is equal to r.
DEFINITION 2. The subsystem whose power
and rank are equal to r is called an rsubsystem.
Following [1], let us introduce number of
concepts, necessary for further reasoning. Let
us examine system P of form (1). We assume
that system P is solvable and has a rank r,
different from zero.
DEFINITION 3. We call the solution of
system P a nodal solution, if it turns into the
equalities some r of its inequalities with the
linearly independent left sides.
DEFINITION 4. The subsystem of system P is
called a nodal subsystem, if its rank is equal to
the number of inequalities in it and all its
nodal solutions satisfy system P.
In [1, chapter 1] is proven the following
theorem.
THEOREM 1.1. Each solvable system of the
linear inequalities of form (1) with rank r,
r > 0, has at least one nodal subsystem, and
hence at least one nodal solution. (And each
nodal subsystem is r-subsystem.)
Let us examine now the inconsistent system S
of form (1). From lemma 1 and theorem 1.1
follows lemma 2.
LEMMA 2. Let system S has a rank r, r > 0.
Then each not extensible solvable subsystem
of system S has at least one nodal rsubsystem.
Therefore for the selection of all not extensible
solvable subsystems of system S it is sufficient
to examine all its r-subsystems. For every of
them it is required to solve the not very
complex task: to find its one nodal solution.
For this purpose it is necessary to replace all
inequality signs in the subsystem by the
equalities signs and to find one solution of the
obtained system of linear equations. The
obtained nodal solution of subsystem must be
substituted into all inequalities of system S and
the inequalities satisfied with this solution
must be selected. The chosen inequalities form
a solvable subsystem. After exhaustion of rsubsystems, we will obtain a certain set which
contains all not extensible solvable subsystems
of system S. Among them it is possible to
select optimum subsystems with the required
properties. Maximum subsystem (according to
the power) among the chosen subsystems is
MSS of system S.
The described method combines finding MSS
with the simultaneous presence of its one
solution that is the final goal of optimization
problem.
However, for the tasks of large dimensionality
complete exhausting of r-subsystems will be
too long. Therefore a number of
approximations methods were developed.
In this work the approximate algorithm A w for
decisions of the formulated problem is
presented. This algorithm realized a partial
directed search of r-subsystems of the assigned
system.
The general work scheme of algorithm Aw
First we transform system S of form (1) as
follows. We normalize the inequalities of the
system so that the modulus of the first nonzero
coefficient in each inequality would be equal
to 1 (signs thus we do not change). Then we
order all inequalities on the increasing of right
sides and afterwards, during construction of rsubsystems, we examine inequalities in this
order.
Algorithm A w has several iterations. At any
iteration we choose the so-called base rsubsystem. Then the set of associated
subsystems by replacing in it some inequalities
is constructed. To the base subsystem and
those from the associated subsystems, which
are r-subsystems, is applied the construction
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procedure of the extended solvable subsystem
(ESS) with the simultaneous presence of one
solution of ESS. Among all obtained ESS
maximum subsystem is selected. The power of
the latter subsystem is compared with value 
– the power of maximum ESS, obtained on the
previous iterations.
As a result after all iterations the record value
of power  and the corresponding solution x
are remembered. By this solution is restored
maximum ESS, which is considered as the
approximation for MSS of system S.
Let us describe now the procedures, mentioned
in the general work scheme of algorithm.
The construction of base r - subsystems
Algorithm A w has a parameter w, which
determines the maximum power of the
intersection of base subsystems. The
correction of parameter w is produced after the
calculation of the rank r of system S. If r = 1
then we assume w = 0, if r = 2 then w = 1, if
w > r-3 (in case of r <n) we assume w = r-3.
Generally at r > 2 values of parameter w can
be in following boundaries: 0  w  r-3.
The first r (in the indicated order) inequalities
with the independent left sides enter into the
1st base subsystem. For constructing the next
base subsystem we move away of system S the
first r-w inequalities, which have entered into
the previous base subsystem. We obtain the
reduced system of linear inequalities, from
which we again select the first r independent
inequalities. So we act until it is possible to
choose an r-subsystem from the reduced
system. The total number b of base subsystems
is estimated as follows:
m  r 
b
 1.
(2)
 r  w 
The construction of the extended solvable
subsystem (ESS)
This procedure can be applied to any rsubsystem of system S. Let the r-subsystem B
is given. We first find the nodal solution of B.
For this we replace all inequality signs in the
subsystem by the equality signs and find the
solution of the obtained system of linear
equations. We find only one solution x (i.e. at
r < n we select one solution of the set). The
obtained solution x we substitute in all
inequalities of system S, we separate those
from them, to which x satisfies. These
inequalities form ESS. We remember the
power  of obtained ESS and the
corresponding solution x.
The construction of the set of subsystems,
associated with the base subsystem
The construction of associated subsystems for
the base subsystem is tree-like process.
Assume that on some iteration a base
subsystem B is built. Let us apply to it the
procedure of construction of ESS. Thus its
nodal solution x will be found, and also
subsystem R(B) - extension of B with power 
is built. Let us consider the set R , which
consists of those inequalities of system S, by
which x does not satisfy.
STAGE 1. First we construct subsystems of
the 1st level, associated with the base rsubsystem B. For this we alternately replace
each of r inequalities of system B by each of
m- inequalities of the set R . We will obtain r
(m- ) associated with B subsystems. We leave
from them only r-subsystems. To each
received r-subsystem Bi we shall apply
procedure of construction of ESS. As a result
the extended solvable subsystem R (Bi) with
power i will be found.
DEFINITION 5. We call an r-subsystem Bi,
associated with the base subsystem B, the
point of increase, if  i > .
STAGE 2. Following cases are possible.
1) Among the subsystems associated with B
there are no points of increase. Then this
branch of process breaks itself and transition
to the next base subsystem is accomplished.
2) The points of increase exist and among
them there is one maximum, i.e., for any rsubsystem Bt will be t >  and t > i for all
i ≠ t. Then for the subsystem Bt the associated
subsystems are constructed exactly as this was
done for the base subsystem B. They are
subsystems of 2nd level.
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3) There are several points of the maximum
increase: subsystem B j1 ,..., B jl , for which will
be  j1  ...   jl . In this case we investigate
alternately these subsystems until we find for
one of them the point of increase.
Then we go over to 3rd stages: it is
constructed, as are above, the subsystems of
3rd level for the maximum point of increase.
And we do not return to the remaining
subsystems of 2nd level.
Let us put the following limitation on number
u of the levels of this process:
 r  w
u
1
(3)
 2 
It is caused by requirement so that the
associated subsystems for the neighboring
base subsystems would not coincide.
Conclusion
After the end of all iterations of algorithm A w
we will obtain the record value  of power of
all constructed ESS and the corresponding
solution x. On this solution is restored the
maximum ESS, which is considered as the
approximation for MSS of system S.
Let us estimate number q of all considered
subsystems of power r. For one examined rsubsystem B it is investigated q(B) such
subsystems:
(4)
q ( B)  r 2 ( m   ) 2 u
Further we have:
r 2 ( m  r )3
, (5)
q  q( B)b  r 2 (m  r )2 bu 
2
where value b is the number of base
subsystems (see formula (2)).
This estimation is overstated: the number of
the subsystems examined will be much less in
practice.
Algorithm A w is tested on tens of tasks. It
works it rapidly and gives a good
approximation for the maximum subsystem of
the system of linear inequalities.
References
1. Chernikov S.N. Linear inequalities. М.:Nauka. 1968.
(In Russian)
2. Katerinochkina N.N. The search methods for the
maximal solvable subsystem of the system of linear
inequalities.
Communication
on
the
applied
mathematics. Computer center of the Russian Academy
of Science. Moscow. 1997. (In Russian)