In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article

Authors

Department of Mathematics‎, ‎Tabriz Branch‎, ‎Islamic Azad University‎, ‎Tabriz‎, ‎Iran‎.

Abstract

This paper presents a capable recurrent neural network, the so-called µRNN for solving a class of non-convex quadratic programming problems‎. ‎Based on the optimality conditions we construct a new recurrent neural network (µRNN)‎, ‎which has a simple structure and its capability is preserved‎. ‎The proposed neural network model is stable in the sense of Lyapunov and converges to the exact optimal solution of the original problem‎. ‎In a particular case‎, ‎the optimality conditions of the problem become necessary and sufficient‎. ‎Numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network.

Keywords

[1] Bacciotti, A. (1992). “Local stabilizability of nonlinear control systems”, Advances in  Mathematics for Applied Sciences.
[2] Bazaraa, M.S., Shetty, C.M. (1990). “Nonlinear programming theory and algorithms”, Wiley and Sons, New York.
[3] Bertsekas, D.P. (1989). “Parallel and distributed numerical methods”, Prentice-Hall, Englewood, Cliffs, NJ.
[4] Best Michael, J. (2017). “Quadratic programming with computer programs”, Advances in Applied Mathematics, University of Waterloo, Canada, CRC Press.
[5] Beyer, D., Ogier, R. (2000). “Tabu learning: A neural networks search method for solving non-convex optimization problems”, IEEE International Joint Conference Neural Networks 2, 953-961.
[6] Bian, W., Chen, X. (2013). “Worst-case complexity of smoothing quadratic regularization methods for non-Lipschitz optimization”, SIAM, Journal of Optimization, 23 (3), 1718-
1741.
[7] Bian, W., Chen, X., Ye, Y. (2014). “Complexity analysis of interior point algorithms for non-Lipschitz and non-convex minimization”, Mathematical Programming, 149(1-2), 301-327.
[8] Boob, D.P. (2020). “Convex and structured non-convex optimization for modern machine learning: Complexity and algorithms”, Georgia Institute of Technology.
[9] Carmon, Y., Duchi, J.C. (2020). “First-order methods for non-convex quadratic minimization”, SIAM Review, 62(2), 395-436.
[10] Chen, X., Womersley, R., Ye, J. (2011). “Minimizing the condition number of a Gram matrix”, SIAM Journal on Optimization, 21, 127-148.
[11] Chicone, C. (2006). “Ordinary differential equations with applications”; Second edition, Springer-Verlag, New York.
[12] Cui, Y., Chang, T. H., Hong, M., Pang, J.S. (2020). “A study of piecewise linear-quadratic programs”, Journal of Optimization Theory and Applications, 1-31.
[13] Effati, S., Mansoori, A., Eshaghnezhad, M. (2015). “A projection neural network for solving bilinear programming problems”, Neurocomputing, 1-20.
[14] Effati, S., Ranjbar, M. (2011). “A novel recurrent nonlinear neural network for solving quadratic programming problems”, Applied Mathematical Modelling, 33, 1688-1695.
[15] Eshaghnezhad, M., Effati, S., Mansoori, A. (2016). “A neurodynamic model to solve non-linear pseudo-monotone projection equation and its applications”, IEEE Transactions on Cybernetics, 1-13.
[16] Gao, X.B., Liao, L.Z., Xue, W. (2004). “A neural network for a class of convex quadratic minimax problems with constraints”, IEEE Transactions on Neural Networks, 15(3), 622-628.
[17] Hopfield, J.J., Tank, D. (1985). “Neural computation of decisions in optimization problem”, Biod. Cybern., 52, 141-152.
[18] Horn, R.A., Johnson, C.R. (1990). “Matrix Analysis”, Cambridge University Press.
[19] Huan, L., Fang, C., Zhouchen, L. (2020). “Accelerated first-order optimization algorithms for machine learning”, Proceedings of the IEEE, doi: 10.1109/JPROC.2020.3007634.
[20] Huan, L., Lin, Z. (2019). “Provable accelerated gradient method for non-convex low-rank optimization”, Machin Learning.
[21] Huang, F., Gu, B., Huo, Z., Xhen, S., Huang, H. (2019). “Faster gradient-free proximal stochastic methods for non-convex nonsmooth optimization”, The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19), 1503-1510.
[22] Jeyakumar, V., Lee, G.M., Li, G.Y. (2009). “Alternative theorems for quadratic inequality systems and global quadratic optimization”, SIAM Journal on Optimization, 20(2), 983-1001.
[23] Jeyakumar, V., Rubinov, A.M., Wu, Z.Y. (2007). “Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions”, Mathematical Programming, series, A 110, 521-541.
[24] Jeyakumar V., Srisatkunarajah S. (2009). “Lagrange multiplier necessary condition for global optimality for non-convex minimization over a quadratic constraint via S-lemma”, Optimization Letters, 3, 23-33.
[25] Khalil, H.K. (2002). “Nonlinear systems”, Prentice Hall, Third edition.
[26] Kong, W., Melo, J.G., Monteiro, R.D.C. (2019). “An efficient adaptive accelerated inexact proximal point method for solving linearly constrained non-convex”, Composite Problems, 1-28.
[27] Leung, M.F., Wang, J. (2019). “Minimax and bi-objective portfolio selection based on collaborative neurodynamic optimization”, IEEE Transactions on Neural Networks and Learning Systems, 1-12.
[28] Liu, S., Jiang, H., Zhang, L., Mei, X. (2020). “A neurodynamic optimization approach for complex-variables programming problem”, Neural Networks, 129, 280-287.
[29] Lu, S. (2018). “First-Order methods of solving non-convex optimization problems: Algorithms, convergence, and optimality”, Electrical and Electronics Commons.
[30] Luenberger, D.G. (1948). “Introduction to Linear and Nonlinear Programming”, Reading MA: Addison-Wesley.
[31] Malek, A., Hosseinipour-Mahani, N. (2015). “Solving a class of non-convex quadratic problems based on generalized KKT conditions and neurodynamic optimization technique”,
Kybernetika, 51, 890-908.
[32] Mansoori, A., Effati, S. (2019). “An efficient neurodynamic model to solve nonlinear programming problems with fuzzy parameters”, Neurocomputing, 334, 125-133.
[33] Mansoori, A., Effati, S. (2019). “Parametric NCP-based recurrent neural network model: A new strategy to solve fuzzy non-convex optimization problems”, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 1-10.
[34] Modarres, F., Hassan, M.A., Leong, W.J. (2011). “A symmetric rank-one method based on extra updating techniques for unconstrained optimization”, Computers and Mathematics with Applications, 62, 392-400.
[35] Nasiri, J., Modarres Khiyabani, F. (2018). “A whale optimization algorithm (WOA)  approach for clustering”, Cogent Mathematics and Statistics, doi.org/10.1080/25742558.2018/1483656.
[36] Nazemi, A.R. (2012). “A dynamic system model for solving convex nonlinear optimization problems”, Communications in Nonlinear Science and Numerical Simulation, 17, 1696-1705.
[37] Nazemi, A.R. (2014). “A neural network model for solving convex quadratic programming problems with some applications problems”, Engineering Applications of Artificial intelligence, 32, 54-62.
[38] Pant, H., Soman Jayadeva, S., Bhaya, A. (2020). “Neurodynamical classifiers with low model complexity”, Neural Networks, 132, 405-415.
[39] Park, S., Jung, S.H., Pardalos, P.M. (2020). “Combining stochastic adaptive cubic regularization with negative curvature for non-convex”, Journal of Optimization Theory and applications, 184 (3), 953-071.
[40] Rudnick-Cohen, E., Herrmann, J.W., Azarm, S. (2020). “Non-convex feasibility robust optimization via scenario generation and local refinement”, Journal of Mechanical Design, 142 (5), 1-10.
[41] Slotine, J.J.E., Li, W. (1990). “Applied Nonlinear Control”, Wiley and Sons, New York.
[42] Strekalovsky, A.S. (2018). “On non-convex optimization problems with D. C. equality and inequality constraints”, IFAC papers online, 51-32, 895-900.
[43] Strekalovsky, A. (2019). “Nonconvex optimization: From global optimality conditions to numerical methods”, AIP Conference Proceedings 2070, 020015, 1-4.
[44] Tank, D.W., Hopfield, J.J. (1986). “Simple neural optimization networks: On A/D converter, signal decision circuit and a linear programming circuit”, IEEE Transactions on Circuits and Systems, 33, 533-541.
[45] Tian, Y., Lu, C. (2011). “Nonconvex quadratic formulations and solvable conditions for mixed integer quadratic programming problems”, Journal of Industrial and Management Optimization, 7 (4), 1027-1039.
[46] Valizadeh Oghani, A., Khiabani, F. M., Farahmand, F.H. (2020). “Data envelopment analysis technique to measure the management ability: Evidence from Iran cement industry”, Cogent Business and Management, doi.org/10.1080/23311975.2020.1801960.
[47] Xu, C., Chai, Y., Qin, S., Wang, Z., Feng, J. (2020). “A neurodynamic approach to nonsmooth constrained pseudo convex optimization problem”, Neural Networks, 124, 180-192.
[48] Xue, X., Bian, W. (2007). “A project neural network for solving degenerate convex quadratic program”, Neurocomputing, 70, 2449-2459.
[49] Yan, Y. (2014). “A new nonlinear neural network for solving quadratic programming problems”, Springer International Publishing Switzerland, 347-357.
[50] Yang, Y., Cao, J., Xu, X., Liu, J. (2012). “A generalized neural network for solving a class of minimax optimization problems with linear constraints”, Applied Mathematics and Computation, 218, 7528-7537.