A feedback neural network for solving convex quadratic bi-level programming problems

In this paper, a feedback neural network model is proposed for solving a class of convex quadratic bi-level programming problems based on the idea of successive approximation. Differing from existing neural network models, the proposed neural network has the least number of state variables and simple structure. Based on Lyapunov theories, we prove that the equilibrium point sequence of the feedback neural network can approximately converge to an optimal solution of the convex quadratic bi-level problem under certain conditions, and the corresponding sequence of the function value approximately converges to the optimal value of the convex quadratic bi-level problem. Simulation experiments on three numerical examples and a portfolio selection problem are provided to show the efficiency and performance of the proposed neural network approach.

[1]  Ue-Pyng Wen,et al.  A neural network approach to multiobjective and multilevel programming problems , 2004 .

[2]  Charles E. Blair,et al.  Computational Difficulties of Bilevel Linear Programming , 1990, Oper. Res..

[3]  Chuandong Li,et al.  Fold-flip bifurcation analysis on a class of discrete-time neural network , 2011, Neural Computing and Applications.

[4]  E. Stanley Lee,et al.  Fuzzy and Multi-Level Decision Making , 2001 .

[5]  Chuandong Li,et al.  Impulsive effects on stability of high-order BAM neural networks with time delays , 2011, Neurocomputing.

[6]  Jinde Cao,et al.  A feedback neural network for solving convex constraint optimization problems , 2008, Appl. Math. Comput..

[7]  Wayne F. Bialas,et al.  On two-level optimization , 1982 .

[8]  Le Dung Muu,et al.  A Global Optimization Method for Solving Convex Quadratic Bilevel Programming Problems , 2003, J. Glob. Optim..

[9]  Jonathan F. Bard,et al.  Practical Bilevel Optimization , 1998 .

[10]  Kwong-Sak Leung,et al.  A new gradient-based neural network for solving linear and quadratic programming problems , 2001, IEEE Trans. Neural Networks.

[11]  Le Thi Hoai An,et al.  A DC programming approach for a class of bilevel programming problems and its application in Portfolio Selection , 2012 .

[12]  Ue-Pyng Wen,et al.  A hybrid neural network approach to bilevel programming problems , 2007, Appl. Math. Lett..

[13]  Zhong Chen,et al.  A neural network for solving a convex quadratic bilevel programming problem , 2010, J. Comput. Appl. Math..

[14]  Huaguang Zhang,et al.  A new result for projection neural networks to solve linear variational inequalities and related optimization problems , 2012, Neural Computing and Applications.

[15]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications) , 2006 .

[16]  Yee Leung,et al.  A high-performance feedback neural network for solving convex nonlinear programming problems , 2003, IEEE Trans. Neural Networks.

[17]  Kwong-Sak Leung,et al.  A Neural Network for Solving Nonlinear Programming Problems , 2002, Neural Computing & Applications.

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

[19]  Francisco Facchinei,et al.  A smoothing method for mathematical programs with equilibrium constraints , 1999, Math. Program..

[20]  Ue-Pyng Wen,et al.  A review of Hopfield neural networks for solving mathematical programming problems , 2009, Eur. J. Oper. Res..

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

[22]  Tiesong Hu,et al.  A neural network approach for solving nonlinear bilevel programming problem , 2008, Comput. Math. Appl..

[23]  J. P. Lasalle.  The stability of dynamical systems , 1976 .

[24]  Z Sheng,et al.  A NEW ALGORITHM BASED ON THE FRANK-WOLFE METHOD AND NEURAL NETWORK FOR A CLASS OF BILEVEL DECISSION MAKING PROBLEMS , 1996 .

[25]  E. Lee,et al.  Fuzzy and Multi-Level Decision Making: An Interactive Computational Approach , 2001 .

[26]  Tingwen Huang,et al.  Stability of Periodic Solution in Fuzzy BAM Neural Networks with Finite Distributed Delays , 2006, ICONIP.

[27]  D. D. Morrison.  Optimization by Least Squares , 1968 .

[28]  Hai Yang,et al.  Models and algorithms for road network design: a review and some new developments , 1998 .

[29]  Jean Bosco Etoa Etoa.  Solving quadratic convex bilevel programming problems using a smoothing method , 2011, Appl. Math. Comput..

[30]  Zhigang Zeng,et al.  Dynamic behaviors of memristor-based delayed recurrent networks , 2012, Neural Computing and Applications.