Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network

IEEE Transactions on Neural Networks

Volumn 17, Issue 6, 2006, Pages 1487-1499

  Hu, Xiaolin ^a   Wang, Jun ^a  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Componentwise pseudomonotone variational inequality; Global asymptotic stability; Projection neural network; Pseudoconvex optimization; Pseudomonotone variational inequality

Indexed keywords

ASYMPTOTIC STABILITY; CONVERGENCE OF NUMERICAL METHODS; OPTIMIZATION; PROBLEM SOLVING;

GLOBAL ASYMPTOTIC STABILITY; PROJECTION NEURAL NETWORK; PSEUDOCONVEX OPTIMIZATION; PSEUDOMONOTONE VARIATIONAL INEQUALITY;

NEURAL NETWORKS;

ALGORITHM; ARTICLE; ARTIFICIAL NEURAL NETWORK; AUTOMATED PATTERN RECOGNITION; INFORMATION RETRIEVAL; METHODOLOGY; SIGNAL PROCESSING;

ALGORITHMS; INFORMATION STORAGE AND RETRIEVAL; NEURAL NETWORKS (COMPUTER); PATTERN RECOGNITION, AUTOMATED; SIGNAL PROCESSING, COMPUTER-ASSISTED;

EID: 34249016393     PISSN: 10459227     EISSN: None     Source Type: Journal    
DOI: 10.1109/TNN.2006.879774     Document Type: Article

References (39)

1. D. Kinderlehrer and G. Stampcchia, An introduction to variational inequalities and their applications. New York: Academic, 1980.
2. R. W. Cottle, F. Giannessi, and J. L. Lions, Variational Inequalities and Complementarity Problems: Theorey and Applications. New York: Wiley, 1980.
3. L. Vandenberghe, B. L. De Moor, and J. Vandewalle, "The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits," IEEE Trans. Circuits Syst., vol. 36, no. 11, pp. 1382-1391, Nov. 1989.
4. P. T. Harker and J. S. Pang, "Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications," Math. Program., vol. 48, pp. 161-220, 1990.
5. T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin, and S. Ganjlizadeh, "Day-to-day dynamic network disequilibria and idealized traveler information systems," Oper. Res., vol. 42, no. 6, pp. 1120-1136, 1994.
6. M. C. Ferris and J. S. Pang, "Engineering and economic applications of complementarity problems," Soc. Ind. Appl. Math. (SIAM) Rev., vol. 39, pp. 669-713, 1997.
7. M. Fukushima, "Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems," Math. Program., vol. 53, pp. 99-110, 1992.
8. T. Larsson and M. Patriksson, "A class of gap functions for variational inequalities," Math. Program., vol. 64, pp. 53-79, 1994.
9. M. V. Solodov and P. Tseng, "Modified projection-type methods for momotone variational inequalities," Soc. Ind. Appl. Math. (SIAM) J. Control Optim., vol. 2, pp. 1814-1830, 1996.
10. B. S. He, Z. H. Yang, and X. M. Yuan, "An approximate proximal-extragradient type method for monotone variational inequalities," J. Math. Anal. Appl., vol. 300, pp. 362-374, 2004.
11. N. E. Farouq, "Pseudomonotone variational inequalities: Convergence of proximal methods," J. Optim. Theory Appl., vol. 109, no. 2, pp. 311-326, May 2001.
12. Y. J. Wang, N. H. Xiu, and C. Y. Wang, "Unified framework of extra-gradient-type methods for pseudomonotone variational inequalities," J. Optim. Theory Appl., vol. 111, no. 3, pp. 641-656, Dec. 2001.
13. M. A. Noor, "Modified projection method for pseudomonotone variational inequalities," Appl. Math. Lett., vol. 15, pp. 315-320, 2002.
14. T. Yoshikawa, Foundations of Robotics: Analysis and Control. Cambridge, MA: MIT Press, 1990.
15. B. Kosko, Neural Networks for Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1992.
16. J. J. Hopfield and D. W. Tank, "'Neural' computation of Decisions in optimization problems," Biol. Cybern., vol. 52, pp. 141-152, 1985.
17. D. W. Tank and J. J. Hopfield, "Simple neural optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit," IEEE Trans. Circuits Syst., vol. CS-33, no. 5, pp. 533-541, May 1986.
18. M. P. Kennedy and L. O. Chua, "Neural networks for nonlinear programming," IEEE Trans. Circuits Syst., vol. 35, no. 5, pp. 554-562, May 1988.
19. A. Rodríguez-Vázquez, R. Domínguez-Castro, A. Rueda, J. L. Huertas, and E. Sánchez-Sinencio, "Nonlinear switched-capacitor neural networks' for optimization problems," IEEE Trans. Circuits Syst., vol. 37, no. 3, pp. 384-397, Mar. 1990.
20. J. Wang, "A deterministic annealing neural network for convex programming," Neural Netw., vol. 7, no. 4, pp. 629-641, 1994.
21. Y. Xia and J. Wang, "A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 51, no. 7, pp. 1385-1394, Jul. 2004.
22. Y. Xia, G. Feng, and J. Wang, "A recurrent neural network with exponential convergence for solving convex quadratic program and linear piecewise equations," Neural Netw., vol. 17, no. 7, pp. 1003-1015, 2004.
23. Y. Xia and J. Wang, "Recurrent neural networks for solving nonlinear convex programs with linear constraints," IEEE Trans. Neural Netw., vol. 16, no. 2, pp. 379-386, Mar. 2005.
24. Y. Xia and J. Wang, "A recurrent neural network for solving linear projection equations," Neural Netw., vol. 13, no. 3, pp. 337-350, 2000.
25. Y. Xia and J. Wang, "On the stability of globally projected dynamical systems," J. Optim. Theory Appl., vol. 106, no. 1, pp. 129-150, 2000.
26. X. Liang and J. Si, "Global exponential stability of neural networks with globally Lipschitz continuous activations and its application to linear variational inequality problem," IEEE Trans. Neural Netw., vol. 12, no. 2, pp. 349-359, Mar. 2001.
27. Y. Xia, H. Leung, and J. Wang, "A projection neural network and its application to constrained optimization problems," IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 49, no. 4, pp. 447-458, Apr. 2002.
28. Y. Xia, "Further results on global convergence and stability of globally projected dynamical systems," J. Optim. Theory Appl., vol. 122, no. 3, pp. 627-649, 2004.
29. M. A. Noor, "A Wiener-Hopf dynamical system for variational inequalities," New Zealand J. Math., vol. 31, pp. 173-182, 2002.
30. X. Hu and J. Wang, "Global stability of a recurrent neural network for solving pseudomonotone variational inequalities," in Proc. IEEE Int. Symp. Circuits Syst., Island of Kos, Greece, May 21-24, 2006, pp. 755-758.
31. M. A. Noor, "Implicit dynamical systems and quasi variational inequalities," Appl. Math. Comput., vol. 134, pp. 69-81, 2003.
32. S. Karamardian and S. Schaible, "Seven kinds of monotone maps," J. Optim. Theory Appl., vol. 66, no. 1, pp. 37-46, 1990.
33. N. Hadjisavvas and S. Schaible, "On strong pseudomonotonicity and (semi)strict quasimonotonicity," J. Optim. Theory Appl., vol. 79, no. 1, pp. 139-155, 1993.
34. R. K. Miller and A. N. Michel, Ordinary Differential Equations. New York: Academic, 1982.
35. M. Avriel, W. E. Diewert, S. Schaible, and I. Zang, Generalized concavity, Mathematical Concepts and Methods in Science and Engineering. New York: Plenum, 1988, vol. 36.
36. A. Cambini, J. P. Crouzeix, and L. Martein, "On the pseudoconvexity of a quadratic fractional function," Optimization, vol. 51, no. 4, pp. 677-687, 2002.
37. M. S. Bazaraa, D. S. Hanifand, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, 2nd ed. New York: Wiley, 1993.
38. 1 equations," J. Oper. Res. Soc. Japan, vol. 29, pp. 352-374, 1986.
39. J. S. Pang and A. Gabriel, "NE/SQP: A robust algorithm for the nonlinear complementarity problem," Math. Program., vol. 60, pp. 295-337, 1993.