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Plenary Lecture
Nonlinear Optimization Models and Solving
Algorithms based on Appropriate Neural Networks
Professor Nicolae Popoviciu
Hyperion University of Bucharest, Romania
Faculty of Mathematics-Informatics
ROMANIA
nicolae.popoviciu@yahoo.com
Abstract:
This work contains a complete set of algorithms for several quadratic and
nonlinear optimization problems. The problem constraints are very
differently. For each type of constraint an appropriate algorithm is given.
The algorithms for linear bound constraints and nonlinear optimization are
based on neural networks and uses a system of differential equations. In
order to reduce the sensitivity and round off errors a preconditioning
method is used. A great number of numerical applications illustrates the
algorithms.
We use
the square matrices M of the type
 or
rectangular matrices M of type
 .
All the
used vectors are column vectors i.e.
 and
denote, for example,
 or
 ,
 .
The letter T means the transposition.
Here we
enumerate several nonlinear optimization models and mention the appropriate
algorithms to solve them.
There
are a lot of quadratic optimization (QO) models (or quadratic programming (QP)
models) and nonlinear optimization models (NO) and here we mention several
of them. We denote by
 the
null vector of an appropriate space, let us say
 and
is
the unknown vector of the any optimization problem.
Model
1. The unconstraint model.
Find
 so
that  .
The unconstrained solution is obtained
from
 .
The matrix
 could
be an invertible or non invertible matrix, but always it is a
symmetric matrix, because we can express
 ,
 ,
(symmetric)
and  (asymmetric),
 .
Model
2. The classical QP model.
Find the
vector so
that  ,
 .
If
 exists,
then the solution
is
obtained by Hildreth D’Esopo algorithm. The algorithm is not based on neural
networks.
Model
3. The QP model with bilateral linear bound constraints].
Find the vector
so
that
,
 ,
 .
This
model is solved by an algorithm based on neural network procedure. The
algorithm has two steps. The first step is a preconditioning technique. The
second step is the solving algorithm.
Model
4. The QP model with one quadratic constraint [8].
Find the
vector so
that  ,
 .
Model
5. The nonlinear convex optimization. (The generalization of
model 2).
Find the
vector x so that
 ,
differentiable, convex;
 .
Model
6. The nonlinear convex optimization, with bilateral linear bound
constraints. (The extension of model 5). Find the vector x so
that ,
F differentiable, convex,
,
 .
Model
7.Variational inequality problem.
Denote
 .
A differentiable vector function
 is
given. Find the vector x* so
that  ,
 .
Now,
shortly we mention that our aim is to solve the model 2 (by Hildreth-D’Esopo
algorithm), model 3 (by preconditioning techniques and Neural Networks) and
models 5,6,7 (by Neural Networks).
Brief Biography of the Speaker:
Name Mr.
Nicolae POPOVICIU
Affiliation Professor Dr. Math.
HYPERION University
of Bucharest
Dean : Faculty of
Math. – Info
Born September 4 , 1943
Place of Born Romania, District of
SIBIU
Nationality Romanian
Education Faculty of Mathematics,
Diploma 1966
University of
Bucharest, Romania
Doctor in Math University of Bucharest,
Diploma 1976
Title Professor ( full )
Place of Job Faculty of Math-Info (
from 2004- today )
Hyperion University of
Bucharest, Romania
Position Dean of Faculty of
Math-Info
Published Books 16
( all in Romanian Language )
Published Papers 83 ( almost all
papers are in English Language )
( 9 papers are in
WSEAS Press, 1 paper in CRC Press )
Plenary Speaker/Chairman Many times Plenary
Speaker and Chairman section
in WSEAS
Conferences
Studies Abroad 1970 ( 9 months
) University Lomonosv of Moscow
1973 ( 6 months )
University Paris VI, France
Visiting Prof 1977
(1 month ) Technical University of Vienna
1978 ( 2 weeks )
Karolin University of Prague
Contact
nicolae.popoviciu@yahoo.com ;
Tel. 0040726 141 266
; 004021 242 89 09
Languages
English, French, Russian
Domains of Interest
1. Probabilities and Statistics.
2. Optimal Strategy for Markov Decision Processes.
Poisson Processes.
3. Distributions and Integral Transforms for
Signal Processing
4. Artificial Neural Networks. Fuzzy Sets and Neural
Networks.
5. Optimization Problems (Linear, Quadratic, Convex,
Nonlinear)
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