Abstract
In this paper, a projection-based recurrent neural network is proposed to solve convex quadratic bilevel programming problems (CQBPP). The Karush–Kuhn–Tucker optimal conditions (KKT) of the lower level problem are used to obtain identical one-level optimization problem. A projected dynamical system which its equilibrium point coincides with the global optimal solution of the corresponding optimization problem is presented. Compared to existing models, the proposed model has the least number of variables and a simple structure with low complexity. Analytically, it is demonstrated that the state vector of the suggested neural network model is stable in the sense of Lyapunov and globally convergent to an optimal solution of CQBPP in finite time. Some numerical examples, a supply chain model and an application deals with an environmental problem are discussed in order to confirm the efficiency of the theoretical results and the performance of the model.
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References
Sinha A, Malo P, Deb K (2017) A review on bilevel optimization: from classical to evolutionary approaches and applications. IEEE Trans Evol Comput 99:1
Dempe S (2002) Foundations of bilevel programming. Springer US, Berlin
Telles E, Lima DA, Contreras J, Alguacil N (2017) A new transmission tariff allocation model based on bilevel programming. IEEE Trans Power Syst 32(3):2204–2213
Dempe S, Kalashnikov V, Perez-Valdes GA, Kalashnykov N (2015) Bilevel programming problems: theory, algorithms and applications to energy networks, 1st edn. Springer, Berlin
Dempe S, Franke S (2016) On the solution of convex bilevel optimization problems. Comput Optim Appl 63:685–703
Qin S, Le X, Wang J (2016) A neurodynamic optimization approach to bilevel quadratic programming. IEEE Trans Neural Netw Learn Syst 28:2580–2591
Rahimi-Gorji M, Ghajar M, Kakaee AH, Ganji DD (2017) Modeling of the air conditions effects on the power and fuel consumption of the SI engine using neural networks and regression. J Braz Soc Mech Sci Eng 39:375–384
Hopfield JJ, Tank DW (1985) Neural computation of decisions in optimization problems. Biol Cybern 52:141–152
Tank DW, Hopfield JJ (1986) Simple neural optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit. IEEE Trans Circuits Syst CS 33:533–541
Malek A, Ezazipour S, Hosseinipour-Mahani N (2011) Projected dynamical systems and optimization problems. Bull Iran Math Soc 37:85–100
Malek A, Hosseinipour-Mahani N, Ezazipour N (2010) Efficient recurrent neural network model for the solution of general nonlinear optimization problems. Optim Methods Softw 25:489–506
Qin S, Liu Y, Xiaoping X, Wang F (2016) A neurodynamic approach to convex optimization problems with general constraint. Neural Netw 84:113–124
Malek A, Yari A (2005) Primal-dual solution for the linear programming problems using neural networks. Appl Math Comput 167:198–211
Hosseini A, Hosseini SM (2013) A new steepest descent differential inclusion-based method for solving general nonsmooth convex optimization problems. J Optim Theory Appl 159:698–720
Liu Q, Wang J (2013) A one-layer projection neural network for nonsmooth optimization subject to linear equalities and bound constraints. IEEE Trans Neural Netw Learn Syst 24:812–824
Hosseinipour-Mahani N, Malek A (2016) A neurodynamic optimization technique based on overestimator and underestimator functions for solving a class of non-convex optimization problems. Math Comput Simul 122:20–34
Hosseinipour-Mahani N, Malek (2015) A Solving a class of non-convex quadratic problems based on generalized KKT conditions and neurodynamic optimization technique. Kybernetika 51:890–908
Sun CY, Feng CB (2005) Neural networks for non-convex nonlinear programming problems: a switching control approach. In: Wang J, Liao X, Yi Z (Eds) Advances in neural networks-ISNN 2005: second international symposium on neural networks, Chongqing, China, May 30-June 1, Proceedings, Part I, Springer, Berlin, pp 694–699
Malek A, Ezazipour S, Hosseinipour-Mahani N (2011) Double projection neural network for solving pseudomonotone variational inequalities. Fixed Point Theory 12:401–418
Xia Y, Wang J (2004) A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans Neural Netw 15:318–328
Golbabai A, Ezazipour S (2017) A high-performance nonlinear dynamic scheme for the solution of equilibrium constrained optimization problems. Expert Syst Appl 82:291–300
Lv Y, Chen Z, Wan Z (2011) A neural network approach for solving mathematical programs with equilibrium constraints. Expert Syst Appl 38:231–234
Qin S, Le X, Wang J (2017) A neurodynamic optimization approach to bilevel quadratic programming. IEEE Trans Neural Netw Learning Syst 28:2580–2591
He X, Chu Li, Huang T, Cha Li, Huang J (2014) A recurrent neural network for solving bilevel linear programming problem. IEEE Trans Neural Netw Learn Syst 5(25):824–830
Li J, Li C, Wu Z, Huang J (2014) A feedback neural network for solving convex quadratic bi-level programming problems. Neural Comput Appl 25:603–611
He X, Chu Li, Huang T, Cha Li (2014) Neural network for solving convex quadratic bilevel programming problems. Neural Netw 51:17–25
Hu T, Guo X, Fu X, Lv Y (2010) A neural network approach for solving linear bilevel programming problem. Knowledge based Syst 23:239–242
Lv Y, Chen Z, Wan Z (2010) A neural network for solving a convex quadratic bilevel programming problem. J Comput Appl Math 234:505–511
Lv Y, Hu T, Wang G, Wan Z (2008) A neural network approach for solving nonlinear bilevel programming problem. Comput Math Appl 55:2823–2829
Lan KM, Wen UP, Shih HS, Lee ES (2007) A hybrid neural network approach to bilevel programming problems. Appl Math Lett 20:880–884
Feng J, Qin S, Shi F, Zhao X (2016) A recurrent neural network with finite-time convergence for convex quadratic bilevel programming problems, Neural Computing and Applications, pp.1-10
Friesz TL, Bernstein DH, Mehta NJ, Toibn RL, Ganjlizadeh D (1994) Day to day dynamical network disequilibria and idealized traveler information systems. J Oper Res 42:1120–1136
Jin L, Li S, Hu B, Liu M (2019) A survey on projection neural networks and their applications. Appl Soft Comput J 76:533–544
John R (2000) A first order characterization of generalized monotonicity. Math Program 88:147–155
Bazaraa MS, Shetty CM (2005) Nonlinear programming theory and algorithms. Wiley, New York
Xia Y (2004) An extended projection neural network for constrained optimization. Neural Comput 16:863–883
Xia Y, Feng G, Wang J (2008) A novel recurrent neural network for solving nonlinear optimization problems with inequality constraints. IEEE Trans Neural Netw 19:1340–1353
Mari R (2014) Integer bilevel linear programming problems: New results and applications, Sapienza University, Ph.D. thesis
Bouzerdoum A, Pattison TR (1993) Neural network for quadratic optimization with bound constraints. IEEE Trans Neural Netw 4:293–304
Miller RK, Michel AN (1982) Ordinary differential equations, 1st edn. Academic Press, New Work
Kuo RJ, Huang CJ (2009) Application of particle swarm optimization algorithm for solving bi-level linear programming problem. Comput Math Appl 58:678–685
Kuo RJ, Han YS (2011) A hybrid of genetic algorithm and particle swarm optimization for solving bi-level linear programming problem. A case study on supply chain model. Appl Math Model 35:3905–3917
Sinha A, Malo P, Deb K, Korhonen P, Wallenius J (2016) Solving bilevel multicriterion optimization problems with lower level decision uncertainty. IEEE Trans Evol Comput 20:199–217
Islam M. M (2018) Development of methods for solving bilevel optimization problems, The University of New South Wales, Australia, Ph.D. thesis
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Golbabai, A., Ezazipour, S. A projection-based recurrent neural network and its application in solving convex quadratic bilevel optimization problems. Neural Comput & Applic 32, 3887–3900 (2020). https://doi.org/10.1007/s00521-019-04391-7
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DOI: https://doi.org/10.1007/s00521-019-04391-7