A Neural Network Approach To Solve Semi-infinite Linear Programming Problems
Volume 5, Issue 3, pp 160 - 166
Publication Date: October 15, 2012
Authors
Alireza Fakharzadeh Fakharzadeh
- Department of Mathematics, Faculty of basic Sciences, Shiraz University of Technology, Shiraz, Iran.
Zahra Alamdar Alamdar
- Department of Mathematics, Faculty of basic Sciences, Shiraz University of Technology, Shiraz, Iran.
Masoumeh Hosseinipour
- Department of Mathematics, Faculty of basic Sciences, Shiraz University of Technology, Shiraz, Iran.
Abstract
In this article, we present a new algorithm for solving Semi-Infinite Linear Programming (SILP) problems based on an artificial neural network concept. First the local reduction method for solving the SILP problems is introduced. Based on the local reduction method, the Karush-Kuhn-Tucker (KKT) conditions and gradient method are used to convert the SILP problem to an unconstrained optimization problem; then, a neural network model is constructed to solve it. Numerical example has been employed to indicate the accuracy of the new method.
Keywords
- Semi-Infinite linear programming
- Neural network
- Local reduction method
- KKT conditions.
References
[1] A. Bouzerdoum and T.R. Pattison, Neural networks for quadratic optimization with bound constraints, IEEE Trans. Neural Networks, 4 (1993), 293-304.
[2] A.V. Fiacco and K. O. Kortanek, Semi-Infinite programming and Application: An Introdction. Symposium, Austin, Texas, September 8-10, 1981.
[3] M.P. Glazos, S. Hui and S.H. Zak, Sliding models in solving convex programming problems, SIAM J. Contr. Optimization, 36 (1998), 680-690.
[4] M.A. Goberna and M.A. Lopez, Linear Semi-infinite Optimization, Alicant University, (1998).
[5] J.J. Hopfield, and D.W. Tunk, Neural computation of decisions in optimization problems, Biol. Cybern., 52 (1985), 141-152.
[6] M.P. Kennedy and L.O. Chua, Neural networks for nonlinear programming, IEEE Trans. Circuits Syst., 35 (1988), 554-562.
[7] T. Leo’n, S. Sanmatias and E. Vercher, On the numerical treatment of linearly constrained semi-infinite optimization problems, European Journal of Operations Research, 121 (2000), 78-91.
[8] P. Mateede, Application of khobotov’s algorithm to variational inequalities and network equilibrium problems, Inform. Syst. Oper. Res., 29 (1991), 258-270.
[9] J.J. Mote and G. Toroaldo, On the solution of large quadratic programming problems with bound constraints, SIAM J. Optimization, 1 (1991), 93-113.
[10] J.S. Pang, A posteriori error bounds for the linearly-constrained variational inequality problem, Math. Operation Res., 12 (1987), 474-484.
[11] R. Reemtsen, and J. Ruckmann, Nonconvex Optimization and Its Application: Semi-Infinite Programming, KLUWER ACADEMIC PUBLISHERS London.
[12] A. Rodriguez-Vazquez, R. Dominguez-Castro, A. Rueda, J.I. Huertas, and E. Sanches-Sinencio, Nonlinear switched-capacitor neural networks for optimization problems, IEEE Trans. Circuits Syst., 37 (1990), 384-397.
[13] W. Sun, and Y. Yuan, Optimization Theory And Methods: Nonlinear Programming, Springer (2006).
[14] 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., CAS-33 (1986), 533-541.
[15] W. Walter, Ordinary Differential Equations ,Graduate text in mathematics, Springer, (1998).
[16] M. Wang, Y. E. Kuo, A perturbation method for solving linear semi-infinite programming problems, Computers and Mathematics with Applications, 37 (1999), 181-198.
[17] X. Wu, J. Xia, J. Li, and W.K. Chen, A high performance neural network for solving linear and quadratic programming problems, IEEE Trans. Neural Networks, 7 (1996), 643-651.
[18] Y. Xia, and Wang, J., Neural network for solving linear programming problems with bounded variables, IEEE Trans. Neural Networks, 6 (1995), 515-519.
[19] Y. Xia, A new Neural network for solving linear programming problems and its applications, IEEE Trans. Neural Networks, 7 (1996), 525-529.
[20] Y. Xia, Anew Neural network for solving linear and quadratic programming problems, IEEE Trans. Neural Networks, 7 (1996), 1544-1547.
[21] Y. Xia, Neural network for solving extended linear programming problems, IEEE Trans. Neural Networks, 8 (1997), 519-525.
[22] Y. Xia, A general methodology for designing globally convergent optimization neural networks, IEEE Trans. Neural Networks, 9 (1998), 1331-1343.