Solving convex programming problems with equality constraints by neural networks

Computers and Mathematics with Applications

Volumn 36, Issue 7, 1998, Pages 41-68

  Chen, Y H ^a   Fang, S C ^b  

^a Fruit of the Loom   (United States)

^b NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Artificial neural networks; Convex programming; Hopfield networks; Penalty function

Indexed keywords

ASYMPTOTIC STABILITY; CONSTRAINT THEORY; LINEAR PROGRAMMING; NUMERICAL METHODS; OPTIMIZATION; PROBLEM SOLVING;

CONVEX PROGRAMMING; HOPFIELD NETWORKS; PENALTY FUNCTION;

NEURAL NETWORKS;

EID: 0032187453     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0898-1221(98)00172-2     Document Type: Article

References (35)

1. L.O. Chua and G.-N. Lin, Nonlinear programming without computation, IEEE Transactions on Circuits and Systems CAS-31 (2), 182-188, (1984).
2. J.J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proceedings of the National Academy of Sciences of the U.S.A. 79, 2554-2558, (1982).
3. J.J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences of the U.S.A. 81, 3088-3092, (1984).
4. J.B. Dennis, Mathematical Programming and Electrical Networks, Chapman and Hall, London, (1959).
5. T.E. Stern, Theory of Nonlinear Networks and Systems, Addison-Wesley, New York, (1965).
6. J. J. Hopfield and D. Tank, Neural computation of decisions in optimization problems, Biological Cybernetics 58, 63-70, (1985).
7. A.G. Tsirukis, G.V. Reklaitis and M.F. Tenorio, Nonlinear optimization using generalized Hopfield networks, Neural Computation 1 (4), 511-521, (1989).
8. B.A. Pearlmutter, Gradient calculations for dynamic recurrent neural networks: A survey, IEEE Transactions on Neural Networks 6 (5), 1212-1228, (1995).
9. S.H. Zak, V. Upatising and S. Hui, Solving linear programming problems with neural networks: A comparative study, IEEE Transactions on Neural Networks 6 (1), 96-104, (1995).
10. C.-Y. Maa and M.A. Shanblatt, Linear and quadratic programming neural network analysis, IEEE Transactions on Neural Networks 3 (4), 580-594, (1992).
11. C.-Y. Maa and M.A. Shanblatt, A two-phase optimization neural network, IEEE Transactions on Neural Networks 3 (6), 1003-1010, (1992).
12. J. Wang, Deterministic annealing neural network for convex programming, Neural Networks 7 (4), 629-641, (1994).
13. M.P. Kennedy and L.-O. Chua, Unifying the Tank and Hopfield linear programming network and the Canonical nonlinear programming circuit of Chua and Lin, IEEE Transactions on Circuits and Systems CAS-34, 210-214, (1987).
14. L.O. Chua and N.N. Wang, Complete stability of autonomous reciprocal nonlinear networks, Circuit Theory and Applications 6, 211-241, (1978).
15. A. Cichocki, R. Unbehauen, K. Weinzierl and R. Hölzel, A new neural network for solving linear programming problems, European Journal of Operational Research 93, 244-256, (1996).
16. M.P. Kennedy and L.-O. Chua, Neural networks for nonlinear programming, IEEE Transactions on Circuits and Systems 35, 554-562, (1988).
17. C.-Y. Maa, C. Chiu and M.A. Shanblatt, A constrained optimization neural net technique for economic power dispatch, 1990 IEEE International Symposium on Circuits and Systems 4, 2946-2950, (1990).
18. W.E. Lillo, S. Hui and S.H. Zak, Neural networks for constrained optimization problems, International Journal of Circuit Theory and Applications 21 (4), 385-399, (1993).
19. W.E. Lillo, M.H. Loh and S. Hui, On solving constrained optimization problems with neural networks: A penalty method approach, IEEE Transactions on Neural Networks 4 (6), 931-940, (1993).
20. D.G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley, MA, (1973).
21. M.S. Bazaraa and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, (1990).
22. S. Barnett and R.G. Cameron, Introduction to Mathematical Control Theory, Second edition, Oxford University Press, New York, (1992).
23. A.M. Liapùnov, Problème général de la stabilitè du mouvement (Reproduction of the French translation in 1907 of a Russian memoire dated 1892), In Annals of Mathematics Studies, Volume 17, Princeton University Press, Princeton, NJ, (1947).
24. J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second edition, revised and expanded, Marcel Dekker, New York, (1994).
25. S. Haykin, Neural Networks, A Comprehensive Foundation, Macmillan, New York, (1994).
26. G.E. Forsythe, M.A. Malcolm and C.B. Moler, Computation Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ, (1977).
27. J. Wang, On the asymptotic properties of recurrent neural networks for optimization, International Journal of Pattern Recognition and Artificial Intelligence 5 (4), 581-601, (1991).
28. J. Wang, A time-varying recurrent neural system for convex programming, In 1991 International Joint Conference on Neural Networks (IJCNN '91, Seattle), pp. 147-152, (1991).
29. J. Wang, Analogue neural network for solving the assignment problem, Electronics Letters 28 (11), 1047-1050, (1992).
30. J. Wang, Analysis and design of a recurrent neural network for linear programming, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 40 (9), 613-618, (1993).
31. S.-C. Fang and S. Puthenpura, Linear Optimization and Extensions: Theory and Algorithms, Prentice-Hall, Englewood Cliffs, NJ, (1993).
32. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, (1970).
33. F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, (1983).
34. S.-C. Fang and S.Y. Wu, An inexact approach to solving linear semi-infinite programming problems, Optimization 28, 291-299, (1994).
35. S.-C. Fang, C.J. Lin and S.Y. Wu, An unconstrained convex programming approach to linear semi-infinite programming, SIAM Journal on Optimization 8 (2), (May 1998).