Neural network models and its application for solving linear and quadratic programming problems

In this paper we consider two recurrent neural network model for solving linear and quadratic programming problems. The first model is derived from an unconstraint minimization reformulation of the program. The second model directly is obtained of optimality condition for an optimization problem. By applying the energy function and the duality gap, we will compare the convergence these models. We also explore the existence and the convergence of the trajectory and stability properties for the neural networks models. Finally, in some numerical examples, the effectiveness of the methods is shown.

[1]  Youshen Xia,et al.  A new neural network for solving linear and quadratic programming problems , 1996, IEEE Trans. Neural Networks.

[2]  A. Fischer.  An NCP–Function and its Use for the Solution of Complementarity Problems , 1995 .

[3]  W K Chen,et al.  A high-performance neural network for solving linear and quadratic programming problems , 1996, IEEE Trans. Neural Networks.

[4]  Jun Wang,et al.  A general methodology for designing globally convergent optimization neural networks , 1998, IEEE Trans. Neural Networks.

[5]  Liqun Qi,et al.  Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations , 1993, Math. Oper. Res..

[6]  Youshen Xia,et al.  Neural network for solving linear programming problems with bounded variables , 1995, IEEE Trans. Neural Networks.

[7]  P. Marcotte.  APPLICATION OF KHOBOTOVS ALGORITHM TO VARIATIONAL INEQUALITIES ANT) NETWORK EQUILIBRIUM PROBLEMS , 1991 .

[8]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[9]  A. Fischer.  A special newton-type optimization method , 1992 .

[10]  Jun Wang,et al.  A recurrent neural network for solving linear projection equations , 2000, Neural Networks.

[11]  Sunil K. Agrawal,et al.  Optimization of Dynamic Systems , 1999 .

[12]  C. Moorehead.  All rights reserved , 1997 .

[13]  Francisco Facchinei,et al.  A semismooth equation approach to the solution of nonlinear complementarity problems , 1996, Math. Program..

[14]  Jong-Shi Pang,et al.  A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem , 1987, Math. Oper. Res..

[15]  Jun Wang,et al.  A projection neural network and its application to constrained optimization problems , 2002 .

[16]  John J. Hopfield,et al.  Simple 'neural' optimization networks: An A/D converter, signal decision circuit, and a linear programming circuit , 1986 .

[17]  Solomon Lefschetz,et al.  Stability by Liapunov's Direct Method With Applications , 1962 .

[18]  Edgar Sanchez-Sinencio,et al.  Nonlinear switched capacitor 'neural' networks for optimization problems , 1990 .

[19]  Francisco Facchinei,et al.  A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems , 1997, Math. Program..

[20]  D. Bertsekas,et al.  TWO-METRIC PROJECTION METHODS FOR CONSTRAINED OPTIMIZATION* , 1984 .

[21]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[22]  Francisco Facchinei,et al.  A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm , 1997, SIAM J. Optim..

[23]  Leon O. Chua,et al.  Nonlinear programming without computation , 1984 .

[24]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[25]  Leon O. Chua,et al.  Neural networks for nonlinear programming , 1988 .

[26]  Jong-Shi Pang,et al.  Nonsmooth Equations: Motivation and Algorithms , 1993, SIAM J. Optim..

[27]  Terry L. Friesz,et al.  Day-To-Day Dynamic Network Disequilibria and Idealized Traveler Information Systems , 1994, Oper. Res..

[28]  Youshen Xia.  A new neural network for solving linear programming problems and its application , 1996, IEEE Trans. Neural Networks.