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\title{Implementable Bounds for a Neural Networks Algorithm in 
Communication Traffic Control}

\author{Alexandru Murgu}

\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ }

\maketitle

\auffil{The author is with Department of Mathematics,
University of Jyvaskyla,
FIN-40351 Jyvaskyla, FINLAND,
Email murgu@julia.math.jyu.fi.}

\begin{abstract}
The traffic control problem for communication networks has been 
widely considered either
from  the standpoint  of traffic assignment, including  
linearizations of  the nonlinear 
traffic  pattern  around the  ``nominal'' regime and variational 
inequalities approach, or 
from  the more classical standpoint of flow networks theory. 
When the traffic  stream is 
time  varying, additional computational  effort is  required, 
and the  problem complexity 
becomes NP-hard, even when we want to solve some of its discrete time 
approximations (it is also NP-hard if we have interval uncertainty in
some of the parameters). 

In this paper,
an  idea from  continuous  bounding  approach often  
encountered in  Linear  Programming 
(deterministic or stochastic)  is  used  to  build  a  
learning  algorithm for the bound
movement  over  the  successive approximation of the true  
solution, trying to  capture a 
``local''  landscape  of a  convex  functional. We implement  the 
learning  algorithm in a 
feedforward neural network (FNN) based on some merit function 
discriminating the quality
of the solution bounds.  Because the traffic pattern (that we want to 
control by  
using this learning 
mechanism) is stochastic, we need to check the convergence 
properties ensuring that 
the  learning algorithm is  asymptotically stable.  For that,
a  suitably chosen 
stochastic Lyapunov function is used.  
Finally, we report some 
simulation  results for a 
Gaussian flow pattern, and we draw the main conclusions of this research. 
\end{abstract}

\begin{thebibliography}{99}

\bibitem{1} R. Zbikowski, P. J. Gawthrop, ``A Survey of Neural Networks for 
Control'', In: K. Warwick, G. W. Irwin, and K. J. Hunt (eds.), {\bf Neural 
      Networks for Control and Systems,} 
Peter Peregrinus Ltd., 1992.

\bibitem{2}  {\it IEEE Control Systems Magazine}, Special Issue on Neural 
Networks in Control Systems,
     April 1990, Vol. 10, No. 3.

\bibitem{3}{\it IEEE Communications Magazine}, Special Issue on Network 
Planning, September 1987, Vol. 25, No. 9.      

\bibitem{4}{\it European  Journal  of  Operational  Research}, Special  
Issue  on Stochastic Control 
     Theory and Operational Research, March 1994, Vol. 73, No. 2.

\bibitem{5}  G. W. Shapiro and H. G. Perros, ``Nested Sliding Window 
Protocols with Packet Fragmentation'', {\it IEEE Trans. on
Communications}, 1993, Vol. 41, No. 1, pp. 99--109.

\bibitem{6}  T. Wang, X. Zhuang, X. Xing and X. Xiao, ``A Neuron-Weighted 
Learning Algorithm  and 
     Its Hardware Implementation in Associative Memories'', {\it IEEE
Trans. on Computers}, 1993, Vol.
     42, No. 5, pp. 636--640.

\bibitem{7}  R. E. Stone and C. A. Tovey,  ``The  Simplex  and  
Projective  Scaling  Algorithms  as  
     Iteratively   Reweighted   Least   Squares  Methods'',  
{\it SIAM Review}, 1991, Vol. 33, No. 2,
     pp. 220--237.

\bibitem{8}  P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, 
``Inertia--Controlling   Methods
     for General Quadratic Programming'', {\it SIAM Review}, 1991, Vol. 33, 
No. 1, pp. 1--36.

\bibitem{9}  P. Tseng, D. P. Bertsekas,  and J. N. Tsitsiklis, ``Partially  
Asynchronous,  Parallel 
     Algorithms for Network Flow and Other Problems'', {\it   SIAM J. 
Control and Optimization},
1990,      Vol. 28, No. 3, pp. 678--710.

\bibitem{10} D. P. Bertsekas and D. El Baz,  ``Distributed  AsynchroNous  
Relaxation  Methods  for 
     Convex  Network  Flow  Problems'', {\it SIAM J. Control and 
Optimization}, 1987, Vol. 25, No. 1,
     pp. 74--85.

\bibitem{11} D. P. Bertsekas  and  S.K. Mitter,  ``A  Descent  Numerical  
Method  for Optimization 
     Problems with Nondifferentiable Cost Functionals'', {\it SIAM J. 
Control}, 1973, Vol. 11, No. 4, 
     pp. 637--652.

\bibitem{12} D. P. Bertsekas, ``An  Auction  Algorithm  for Shortest Paths'',
{\it SIAM J. Optimization}, 1991, 
     Vol. 1, No. 4, pp. 425--447.

\bibitem{13} E. Fossas and J.M. Olm,  ``Generation  of  Signals  in a Buck 
Converter With Sliding 
     Mode Control'', {\it ISCAS'94, Proc. of the 1994 IEEE International 
Symposium on Circuits
     and Systems,  Vol. 6,  Nonlinear  Circuits  \&  Systems}, 
Neural Systems, London, 1994,  pp.
     157--160.

\bibitem{14} A. Murgu, P. Neittaanmaki, and V. Hara,  ``A Neural Networks 
Approach of Routing/Flow
     Control for Communication Networks'', in  {\it Proceedings of 
ICNN'94, IEEE International
     Conference on Neural Networks}, 1994, pp. 2667--2672.

\bibitem{15} A. Murgu,  P. Neittaanmaki,  and  V. Hara,  ``A  Neural  
Networks Approach of Dynamic 
     Priority Assignment in a Queueing Network'', in: 
P. A. S. Ralston and T. L. Ward (eds.),
{\it Proceedings of the 1994 International
     Fuzzy Systems and Intelligent Control Conference}, 1994,
     pp. 248--257.

\end{thebibliography}

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