A projection neural-dynamic model for solving fuzzy convex nonlinear programming problems

Jahangiri, Mohammadreza; Nazemi, Alireza

doi:10.22111/ijfs.2024.46972.8276

Document Type : Research Paper

Authors

¹ Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Tel-Fax No:+98-23-32300235, Shahrood, Iran.

² Shahrood University of Technology

https://doi.org/10.22111/ijfs.2024.46972.8276

Abstract

In the proposed manuscript, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is gained
using a projection recurrent neural network (RNN) scheme. Since there is a few research for resolving of FNLOP
by RNN's, we establish a new scheme to solve the problem. By reducing the
original program to an interval problem and then weighting problem, the Karush--Kuhn--Tucker (KKT)
conditions are presented. Moreover, we apply the KKT conditions into a RNN as a efficient tool to solve the problem. Besides, the convergence properties and the
stability analysis of the system model are provided. In the final step, several simulation examples are verified to support the obtained results. Reported results are compared with some other previous neural networks.

Keywords

Main Subjects

Optimization

References

[1] S. K. Agrawal, B. C. Fabien, Optimization of dynamic systems, Kluwer Academic Publishers, Netherlands, 1999.
https://doi.org/10.1007/978-94-015-9149-2

[2] Z. Arjmandzadeh, M. Safi, A. Nazemi, A new neural network model for solving random interval linear programming
problems, Neural Networks, 89 (2017), 11-18. https://doi.org/10.1016/j.neunet.2016.12.007

[3] M. Avriel, Nonlinear programming: Analysis and methods, Prentice-Hall, EnglewoodCliffs, NJ, 1976.

[4] M. S. Bazaraa, C. Shetty, H. D. Sherali, Nonlinear programming, theory and algorithms, John Wiley and Sons,
NewYork, 1979. https://doi.org/10.1002/0471787779

[5] D. P. Bertsekas, J. N. Tsitsiklis, Parallel and distributed computation: Numerical methods, Prentice Hall. New
Jersey: Englewood Cliffs, 1989.

[6] S. Boyd, L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, 2004. https://doi.
org/10.1109/TAC.2006.884922

[7] J. J. Buckley, Possibilistic linear programming with triangular fuzzy numbers, Fuzzy Sets and Systems, 26 (1998),
135-138. https://doi.org/10.1016/0165-0114(88)90013-9

[8] L. Campos, J. L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems,
32 (1989), 1-11. https://doi.org/10.1016/0165-0114(89)90084-5

[9] X. Cao, A. Francis, X. Pu, Z. Zhang, V. Katsikis, P. Stanimirovic, I. Brajevic, S. Li, A novel recurrent neural
network based online portfolio analysis for high frequency trading, Expert Systems with Applications, 233 (2023),
120934. https://doi.org/10.1016/j.eswa.2023.120934

[10] X. Cao, C. Peng, Y. Zheng, S. Li, T. T. Ha, V. Shutyaev, V. Katsikis, P. Stanimirovic, Neural networks for
portfolio analysis in high-frequency trading, IEEE Transactions on Neural Networks and Learning Systems, 2023.
https://doi.org/10.1109/TNNLS.2023.3311169

[11] Y. H. Chen, S. C. Fang, Neurocomputing with time delay analysis for solving convex quadratic programming
problems, IEEE Transactions on Neural Networks, 11 (2000), 230-240. https://doi.org/10.1109/72.822526

[12] J. S. Chen, C. H. Ko, S. Pan, A neural network based on the generalized Fischer-Burmeister function for nonlinear
complementarity problems, Information Sciences, 180(5) (2010), 697-711. https://doi.org/10.1016/j.ins.
2009.11.014

[13] S. Effati, A. Ghomashi, A. Nazemi, Application of projection neural network in solving convex programming
problems, Applied Mathematics and Computation, 188 (2007), 1103-1114. https://doi.org/10.1016/j.amc.
2006.10.088

[14] S. Effati, A. Mansoori, M. Eshaghnezhad, An efficient projection neural network for solving bilinear programming
problems, Neurocomputing, 168 (2015), 1188-1197. https://doi.org/10.1016/j.neucom.2015.05.003

[15] M. Ehrgott, M. Wiecek, Multiobjective programming, In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multicriteria
Decision Analysis: State of the Art Surveys., Springer Science + Business Media, New York, (2005), 667-722.
https://doi.org/10.1007/b100605

[16] M. Eshaghnezhad, S. Effati, A. Mansoori, A neurodynamic model to solve nonlinear pseudo-monotone projection
equationand its applications, IEEE Transactions on Cybernetics, 47(10) (2017), 3050-3062. https://doi.org/
10.1109/TCYB.2016.2611529

[17] M. A. Fatma, A differential equation approach to fuzzy non-linear programming problems, Fuzzy Sets and Systems,
93 (1998), 57-61. https://doi.org/10.1016/S0165-0114(96)00217-5

[18] A. Feizi, A. Nazemi. A. Feizi, A. Nazemi, M. Rabiei, Solving the stochastic support vector regression with probabilistic constraints by a high-performance neural network model, Engineering with Computers, 38 (2021), 1-16.
https://doi.org/10.1007/s00366-020-01214-5

[19] R. Fletcher, Practical methods of optimization, Wiley, NewYork, 1981. https://doi.org/10.1002/
9781118723203

[20] M. Friedman, M. Ma, A. Kandel, Numerical solution of fuzzy differential and integral equations, Fuzzy Sets and
Systems, 106 (1999), 35-48. https://doi.org/10.1016/S0165-0114(98)00355-8

[21] T. L. Friesz, D. H. Bernstein, N. J. Mehta, R. L. Tobin, S. Ganjlizadeh, Day-to-day dynamic network disequilibria
and idealized traveler information systems, Operation Research, 42 (1994), 1120-1136. https://doi.org/10.
1287/opre.42.6.1120

[22] X. Gao, A novel neural network for nonlinear convex programming, IEEE Transactions on Neural Networks, 15
(2004), 613-621. https://doi.org/10.1109/TNN.2004.824425

[23] R. Ghanbari, K. Ghorbani-Moghadam, N. Mahdavi-Amiri, A variables neighborhood search algorithm for solving
fuzzy quadratic programming problems using modified Kerre’s method, Soft Computing, 23 (2019), 12305-12315.
https://doi.org/10.1007/s00500-019-03771-4

[24] R. Ghanbari, K. Ghorbani-Moghadam, N. Mahdavi-Amiri, A time variant multi-objective particle swarm optimization algorithm for solving fuzzy number linear programming problems using modified Kerre’s method, OPSEARCH,
58 (2021), 403-424. https://doi.org/10.1007/s12597-020-00482-5

[25] X. He, T. Huang, J. Yu, C. Li, C. Li, An inertial projection neural network for solving variational inequalities,
IEEE Transactions on Cybernetics, 47 (2016), 809-814. https://doi.org/10.1109/TCYB.2016.2523541

[26] J. J. Hopfield, D. W. Tank, Neural computation of decisions in optimization problems, Biological Cybernetics, 52
(1985), 141-152. https://doi.org/10.1007/BF00339943

[27] C. Y. Hsu, B. R. Kao, V. L. Hob, K. R. Lai, Agent-based fuzzy constraint-directed negotiation mechanism for
distributed job shop scheduling, Engineering Applications of Artificial Intelligence, 53 (2016), 140-154. https:
//doi.org/10.1016/j.engappai.2016.04.005

[28] X. Hu, J. Wang, Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using
the projection neural network, IEEE Transactions on Neural Networks, 17 (2006), 1487-1499. https://doi.org/
10.1109/TNN.2006.879774

[29] D. Karbasi, A. Nazemi, M. Rabiei, A parametric recurrent neural network scheme for solving a class of fuzzy
regression models with some real-world applications, Soft Computing, 24 (2020), 11159-11187. https://doi.
org/10.1007/s00500-020-05008-1

[30] D. Karbasi, A. Nazemi, M. Rabiei, An optimization technique for solving a class of ridge fuzzy regression problems,
Neural Processing Letters, 53 (2021), 3307-3338. https://doi.org/10.1007/s11063-021-10538-2

[31] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Society for
Industrial and Applied Mathematics, 2000. https://doi.org/10.1137/1.9780898719451

[32] S. Li, J. He, Y. Li, M. U. Rafique, Distributed recurrent neural networks for cooperative control of manipulators: A
game-theoretic perspective, IEEE Transactions on Neural Networks and Learning Systems, 28(2) (2016), 415-426.
https://doi.org/10.1109/TNNLS.2016.2516565

[33] S. Li, L. Jin, M. A. Mirza, Kinematic control of redundant robot arms using neural networks, John Wiley and
Sons, 2019. http://dx.doi.org/10.1002/9781119557005

[34] S. Li, Y. Li, Z. Wang, A class of finite-time dual neural networks for solving quadratic programming problems
and its k-winners-take-all application, Neural Networks, 39 (2013), 27-39. https://doi.org/10.1016/j.neunet.
2012.12.009

[35] Z. Li, H. Xiao, C. Yang, Y. Zhao, Model predictive control of nonholonomic chained systems using general projection neural networks optimization, IEEE Transactions on Systems, Man, and Cybernetics, 45 (2015), 1313-1321.
https://doi.org/10.1109/TSMC.2015.2398833

[36] G. Li, Z. Yan, J. Wang, A one-layer recurrent neural network for constrained nonconvex optimization, Neural
Networks, 61 (2015), 10-21. https://doi.org/10.1016/j.neunet.2014.09.009

[37] L. Z. Liao, H. Qi, L. Qi, Solving nonlinear complementarity problems with neural networks: A reformulation
method approach, Journal of Computational and Applied Mathematics, 131(1-2) (2001), 343-359. https://doi.
org/10.1016/S0377-0427(00)00262-4

[38] Q. Liu, J. Wang, L1-minimization algorithms for sparse signal reconstruction based on a projection neural network,
IEEE Transactions on Neural Networks and Learning Systems, 27(3) (2016), 698-707. https://doi.org/10.
1109/TNNLS.2015.2481006

[39] J. Lu, S. C. Fang, Solving nonlinear optimization problems with fuzzy relation equation constraints, Fuzzy Sets
and Systems, 119 (2001), 1-20. https://doi.org/10.1016/S0165-0114(98)00471-0

[40] T. Lu, S. T. Liu, Fuzzy nonlinear programming approach to the evaluation of manufacturing processes, Engineering
Applications of Artificial Intelligence, 72 (2018), 183-189. https://doi.org/10.1016/j.engappai.2018.04.003

[41] K. Majeed, Z. Masood, R. Samar, M. A. Z. Raja, A genetic algorithm optimized Morlet wavelet artificial neural
network to study the dynamics of nonlinear Troesch’s system, Applied Soft Computing, 56 (2017), 420-435.
https://doi.org/10.1016/j.asoc.2017.03.028

[42] A. Mansoori, S. Effati, An efficient neurodynamic model to solve nonlinear programming problems with fuzzy
parameters, Neurocomput, 334 (2019), 125-133. https://doi.org/10.1016/j.neucom.2019.01.012

[43] A. Mansoori, S. Effati, M. Eshaghnezhad, An efficient recurrent neural network model for solving fuzzy
non-linear programming problems, Applied Intelligence, 46 (2017), 308-327. https://doi.org/10.1007/
s10489-016-0837-4

[44] A. Mansoori, S. Effati, M. Eshaghnezhad, A neural network to solve quadratic programming problems with
fuzzy parameters, Fuzzy Optimization and Decision Making, 17(1) (2018), 75-101. https://doi.org/10.1007/
s10700-016-9261-9

[45] A. Mansoori, M. Erfanian, A dynamic model to solve the absolute value equations, Journal of Computational and
Applied Mathematics, 333 (2018), 28-35. https://doi.org/10.1016/j.cam.2017.09.032

[46] A. Mansoori, M. Eshaghnezhad, S. Effati, An efficient neural network model for solving the absolute value
equations, IEEE Transactions on Circuits and Systems II: Express Briefs, 65(3) (2018), 391-395. https:
//doi.org/10.1109/TCSII.2017.2750065

[47] X. Miao, J. S. Chen, C. H. Ko, A smoothed NR neural network for solving nonlinear convex programs with secondorder cone constraints, Information Sciences, 268(1) (2014), 255-270. https://doi.org/10.1016/j.ins.2013.
10.017

[48] K. M. Miettinen, Non-linear multi objective optimization, Kluwer Academic Publishers, 1999. https://doi.org/
10.1007/978-1-4615-5563-6

[49] R. K. Miller, A. N. Michel, Ordinary differential equations, Academic Press, NewYork, 1982. http://dx.doi.
org/10.1109/MCS.2008.930834

[50] S. M. Mousavi, S. Mirdamadi, A. Siadat, J. Dantan, R. Tavakkoli-Moghaddam, An intuitionistic fuzzy grey
model for selection problems with an application to the inspection planningin manufacturing firms, Engineering
Applications of Artificial Intelligence, 39 (2015), 157-167. https://doi.org/10.1016/j.engappai.2014.12.004

[51] A. R. Nazemi, A dynamic system model for solving convex nonlinear optimization problems, Communications in
Nonlinear Science and Numerical Simulation, 17 (2012), 1696-1705. https://doi.org/10.1016/j.cnsns.2011.
08.035

[52] A. R. Nazemi, S. Effati, Anapplication of a merit function for solving convex programming problems, Computers
and Industrial Engineering, 66 (2013), 212-221. https://doi.org/10.1016/j.cie.2013.07.017

[53] A. R. Nazemi, M. Mortezaee, A new gradient-based neural dynamic framework for solving constrained min-max
optimization problems with an application in portfolio selection models, Applied Intelligence, 49 (2019), 396-419.
https://doi.org/10.1007/s10489-018-1268-1

[54] A. R. Nazemi, A. Sabeghi, A new neural network framework for solving convex secondorder cone constrained
variational inequality problems with an application in multi-finger robot hands, Journal of Experimental and
Theoretical Artificial Intelligence, 32(2) (2019), 181-203. https://doi.org/10.1080/0952813X.2019.1647559

[55] A. Nikseresht, A. R. Nazemi, A novel neural network for solving semidefinite programming problems with some
applications, Journal of Computational and Applied Mathematics, 350 (2019), 309-323. https://doi.org/10.
1016/j.cam.2018.10.025

[56] A. Nikseresht, A. R. Nazemi, A novel neural network model for solving a class of nonlinear semidefinite programming problems, Journal of Computational and Applied Mathematics, 338 (2018), 69-79. https://doi.org/10.
1016/j.cam.2018.01.023

[57] A. Nikseresht, A. R. Nazemi, A smoothing gradient-based neural network strategy for solving semidefinite programming problems, NET-Network: Computation in Neural Systems, 33(3-4) (2022), 187-213. https://doi.org/10.
1080/0954898X.2022.2104463

[58] J. Nocedal, S. Wright, Numerical optimization, Second ed., Springer-Verlag, Berlin, NewYork, 2006. https:
//doi.org/10.1007/0-387-22742-3_18

[59] Z. Peng, J. Wang, Output-feedback path-following control of autonomous under water vehicles based on an extended state observer and projection neural networks, IEEE Transactions on Systems, Man, and Cybernetics, 48(4)
(2018), 535-544. https://doi.org/10.1109/TSMC.2017.2697447

[60] S. Qin, X. Le, J. Wang, A neurodynamic optimization approach to bilevel quadratic programming, IEEE Transactions on Neural Networks and Learning Systems, 28 (2017), 2580-2591. https://doi.org/10.1109/TNNLS.
2016.2595489

[61] M. A. Z. Raja, M. A. Manzar, F. H. Shah, Intelligent computing for Mathieu’s systems for parameter excitation,
vertically driven pendulum and dusty plasma models, Applied Soft Computing, 62 (2018), 359-372. https://doi.
org/10.1016/j.asoc.2017.10.049

[62] D. Rani, A. Ebrahimnejad, G. Gupta, Generalized techniques for solving intuitionistic fuzzy multi-objective nonlinear optimization problems, Expert Systems with Applications, 202 (2022), 117264. https://doi.org/10.
1016/j.eswa.2022.117264

[63] M. Ranjbar, S. Effati, S. M. Miri, An artificial neural network for solving quadratic zero-one programming problems, Neurocomputing, 235 (2017), 192-198. https://doi.org/10.1016/j.neucom.2016.12.064

[64] S. Rao, Engineering optimization: Theory and practice, Fourth ed., John Wiley and Sons, Hoboken, NewJersey,
2009. https://doi.org/10.1002/9780470549124

[65] J. D. L. C. Sama, K. Some, Solving fuzzy nonlinear optimization problems using null set concept, International
Journal of Fuzzy System, 26 (2023), 674-685. https://doi.org/10.1007/s40815-023-01626-7

[66] Y. Shi, W. Sheng, S. Li, B. Li, X. Sun, Neurodynamics for equality-constrained time-variant nonlinear optimization
using discretization, IEEE Transactions on Industrial Informatics, 20 (2024), 2354-2364. https://doi.org/10.
1109/TII.2023.3290187

[67] Y. Shi, Q. Shi, X. Cao, B. Li, X. Sun, An advanced discrete-time RNN for handling discrete time-varying matrix
inversion: Form model design to disturbance-suppression analysis, CAAI Transactions on Intelligence Technology,
(2023), 607-621. https://doi.org/10.1049/cit2.12229

[68] Y. Shi, J. Wang, S. Li, B. Li, X. Sun, Tracking control of cable-driven planar robot based on discrete-time recurrent
neural network with immediate discretization method, IEEE Transactions on Industrial Informatics, 19 (2023),
7414-7423. https://doi.org/10.1109/TII.2022.3210255

[69] Y. Shi, W. Zhao, S. Li, B. Li, X. Sun, Novel discrete-time recurrent neural network for robot manipulator: A
direct discretization technical route, IEEE Transactions on Neural Networks and Learning Systems, 34 (2023),
2781-2790. https://doi.org/10.1109/TNNLS.2021.3108050

[70] G. Singh, A. Singh, Extension of particle swarm optimization algorithm for solving transportation problem in
fuzzy environment, Applied Soft Computing, 110 (2021), 107619. https://https://doi.org/10.1016/j.asoc.
2021.107619

[71] L. Stefanini, L. Sorini, M. L. Guerra, Parametric representation of fuzzy numbers and application to fuzzy calculus,
Fuzzy Sets and Systems, 157 (2006), 2423-2455. https://doi.org/10.1016/j.fss.2006.02.002

[72] J. Sun, J. S. Chen, C. H. Ko, Neural networks for solving second-order cone constrained variational inequality problem, Computational Optimization and Applications, 51 (2012), 623-648. https://doi.org/10.1007/
s10589-010-9359-x

[73] J. Sun, L. Zhang, A globally convergent method based on Fischer-Burmeister operators for solving second-order
coneconstrained variational inequality problems, Computational Optimization and Applications, 58 (2009), 1936-
1946. https://doi.org/10.1016/j.camwa.2009.07.084

[74] Y. R. Syau, On convex and concave fuzzy mappings, Fuzzy Sets and Systems, 103 (1999), 163-168. https:
//doi.org/10.1016/S0165-0114(97)00210-8

[75] J. Tang, D. Wang, Modelling and optimization for a type of fuzzy nonlinear programming problems in manufacturing systems, in: Proceedings of the 35th IEEE Conference on Decision and Control, (1996), 4401-4405.
https://doi.org/10.1109/CDC.1996.577485.

[76] D. W. Tank, J. J. Hopfield, Simple neural optimization networks: An a/d converter, signal decisioncircuit, and a
linear programming circuit, IEEE Transactions on Circuits and Systems, 33 (1986), 533-541. https://doi.org/
10.1109/TCS.1986.1085953

[77] P. Vasant, Hybrid LS-SA-PS methods for solving fuzzy non-linear programming problems, Mathematical and
Computer Modelling, 57 (2013), 180-188. https://doi.org/10.1016/j.mcm.2011.08.002

[78] G. Wang, C. Wu, Directional derivatives and sub-differential of convex fuzzy mappings and application in convex
fuzzy programming, Fuzzy Sets and Systems, 138 (2003), 559-591. https://doi.org/10.1016/S0165-0114(02)
00440-2

[79] H. C. Wu, Evaluate fuzzy optimization problems based on biobjective programming problems, Computers and
Mathematics with Applications, 47 (2004), 893-902. https://doi.org/10.1016/S0898-1221(04)90073-9

[80] C. W. Wu, M. Y. Liao, Fuzzy nonlinear programming approach for evaluating and ranking process yields with
imprecise data, Fuzzy Sets and Systems, 246 (2014), 142-155. https://doi.org/10.1016/j.fss.2013.10.014

[81] Y. Xia, An extended neural network for constrained optimization, Neural Computation, 16 (2004), 863-883.
https://doi.org/10.1162/089976604322860730

[82] Y. Xia, J. Wang, A recurrent neural network for solving linear projection equations, Neural Networks, 13 (2000),
337-350. https://doi.org/10.1016/S0893-6080(00)00019-8

[83] Y. Xia, J. Wang, A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality
constraints, IEEE Transactions on Circuits and Systems I: Regular Papers, 51(7) (2004), 1385-1394. https:
//doi.org/10.1109/TCSI.2004.830694

[84] Y. Xia, J. Wang, A bi-projection neural network for solving constrained quadratic optimization problems, IEEE
Transactions on Neural Networks and Learning Systems, 27(2) (2015), 214-224. https://doi.org/10.1109/
TNNLS.2015.2500618

[85] Y. Yang, X. Xu, The projection neural network for solving convex nonlinear programming, In: Huang D-S, Heutte
L, Loog M (eds) ICIC 2007, LNAI, vol 4682. Springer-Verlag, Heidelberg, (2007), 174-181. https://doi.org/
10.1007/978-3-540-74205-0_20

[86] J. Zhang, X. Chen, L. Li, X. Ma, Optimality conditions for fuzzy optimization problems under granular convexity
concept, Fuzzy Sets and Systems, 447 (2022), 54-75. https://doi.org/10.1016/j.fss.2022.01.004

[87] H. Zhu, G. H. Huang, P. Guo, SIFNP: Simulation–based interval–fuzzy nonlinear programming for seasonal
planning of stream water quality management, Water, Air, and Soil Pollution, 223 (2012), 2051-2072. https:
//doi.org/10.1007/s11270-011-1004-5

[88] H. Zhu, G. H. Huang, P. Guo, X. S. Qin, A fuzzy robust nonlinear programming model for stream water quality management, Water Resources Management, 23 (2009), 2913-2940. https://doi.org/10.1007/
s11269-009-9416-3

Iranian Journal of Fuzzy Systems

A projection neural-dynamic model for solving fuzzy convex nonlinear programming problems

References

References

Volume 21, Issue 3
May and June 2024
Pages 37-63

A projection neural-dynamic model for solving fuzzy convex nonlinear programming problems

References

References

Volume 21, Issue 3May and June 2024Pages 37-63

Volume 21, Issue 3
May and June 2024
Pages 37-63