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A One-Layer Recurrent Neural Network for Interval-Valued Optimization Problem with Linear Constraints

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Abstract

In this paper, the interval-valued optimization problem is converted to a general problem in the parametric form and its solution is efficient. We present a one-layer recurrent neural network for solving this interval-valued optimization problem with linear constraints. Based on this approach, we prove that the recurrent neural network is stable in the sense of Lyapunov and the equilibrium point of the neural network is globally convergent to the optimal solution. The proposed approach improves the algorithm for the interval-valued optimization and the model is easy to implement. Finally, two numerical examples are provided to show the feasibility and effectiveness of the proposed approach.

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References

  1. Soyster AL (1979) Inexact linear programming with generalized resource sets. Eur J Oper Res 3(4):316–321

    Article  MathSciNet  Google Scholar 

  2. Thuente DJ (1980) Duality theory for generalized linear programs with computational methods. Oper Res 28(4):1005–1011

    Article  MathSciNet  Google Scholar 

  3. Wu HC (2011) Interval-valued optimization problems based on different solution concepts. Pac J Optim 7(1):173–193

    MathSciNet  MATH  Google Scholar 

  4. Ghosh D (2017) Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions. J Appl Math Comput

  5. Su ZG, Wang PH (2015) Parameter estimation from interval-valued data using the expectation-maximization algorithm. J Stat Comput Simul 85(1–3):320–338

    Article  MathSciNet  Google Scholar 

  6. Wu HC (2007) The Karush–Kuhn–Tucker optimality conditions in an optimization problem with interval-valued objective function. Eur J Oper Res 176(1):46–59

    Article  MathSciNet  Google Scholar 

  7. Chen S (2020) The KKT optimality conditions for optimization problem with interval-valued objective function on Hadamard manifolds. Optimization 1:1–20

    Google Scholar 

  8. Bhurjee AK, Panda G (2012) Efficient solution of interval optimization problem. Math Methods Oper Res 76(3):273–288

    Article  MathSciNet  Google Scholar 

  9. Chalco-Cano Y (2015) A note on optimality conditions to interval optimization problems. In: 2015 conference of the international fuzzy systems association and the European society for fuzzy logic and technology (IFSA-EUSFLAT-15)

  10. Sun J, Miao Z, Gong D (2019) Interval multiobjective optimization with memetic algorithms. IEEE Trans Cybern 48(99):1–14

    Article  Google Scholar 

  11. Wang L, Chen Z, Yang G (2020) An interval uncertain optimization method using back-propagation neural network differentiation. Comput Methods Appl Mech Eng 366

  12. Tao Q, Xin L, Cui X (2005) A linear optimization neural network for associative memory. Appl Math Comput 171(2):1119–1128

    MathSciNet  MATH  Google Scholar 

  13. Guo Z, Baruah SK (2016) A neurodynamic approach for real-time scheduling via maximizing piecewise linear utility. IEEE Trans Neural Netw Learn Syst 27(2):238–248

    Article  MathSciNet  Google Scholar 

  14. Yang Y, Guo Z, Xiong H, Ding DW (2019) Data-driven robust control of discrete-time uncertain linear systems via off-policy reinforcement learning. IEEE Trans Neural Netw Learn Syst 30(12):3735–3747

    Article  MathSciNet  Google Scholar 

  15. Hu X, Wang J (2007) Design of general projection neural networks for solving monotone linear variational inequalities and linear and quadratic optimization problems. IEEE Trans Syst Man Cybern Part B Cybern A Publ IEEE Syst Man Cybern Soc 37(5):1414–1421

    Article  Google Scholar 

  16. 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 33(5):533–541

    Article  Google Scholar 

  17. Kennedy MP, Chua LO (1988) Neural networks for nonlinear programming. IEEE Trans Circuits Syst 33(5):554–562

    Article  MathSciNet  Google Scholar 

  18. Xia Y, Wang J (2005) A recurrent neural network for solving nonlinear convex programs subject to linear constraints. IEEE Trans Neural Netw 16(2):379–386

    Article  Google Scholar 

  19. Xue X, Bian W (2008) Subgradient-based neural networks for nonsmooth convex optimization Problems. IEEE Trans Circuits Syst I Regular Pap 55(8):2378–2391

    Article  MathSciNet  Google Scholar 

  20. Xue X, Bian W (2009) Subgradient-based neural networks for nonsmooth convex optimization problems. IEEE Trans Neural Netw 20(6):1024–10381

    Article  Google Scholar 

  21. Arjmandzadeh Z, Safi (2017) A new neural network model for solving random interval linear programming problems. Neural Netw 89, 11

  22. Nikseresht A, Nazemi A (2018) A novel neural network model for solving a class of nonlinear semidefinite programming problems. J Comput Appl Math 338:69–79

    Article  MathSciNet  Google Scholar 

  23. Li W, Bian W, Xue X (2020) Projected neural network for a class of non-Lipschitz optimization problems with linear constraints. IEEE Trans Neural Netw Learn Syst 31(9):3361–3373

    Article  MathSciNet  Google Scholar 

  24. Liu, Q, Wang J (2008) A one-layer recurrent neural network for convex programming. In: IEEE International joint conference on neural networks (IEEE world congress on computational intelligence)

  25. Liu Q, Wang J (2011) A one-layer recurrent neural network for constrained single-ratio linear fractional programming. In: IEEE international symposium on circuits & systems

  26. Qin S, Yang X, Xue X (2017) A one-layer recurrent neural network for pseudoconvex optimization problems with equality and inequality constraints. IEEE Trans Cybern 47(10):3063–3074

    Article  Google Scholar 

  27. Liu Q, Guo Z, Wang J (2012) A one-layer recurrent neural network for constrained pseudoconvex optimization and its application for dynamic portfolio optimization. Neural Netw 26:99–109

    Article  Google Scholar 

  28. Guo Z, Liu Q, Wang J (2014) A simplified recurrent neural network for pseudoconvex optimization subject to linear equality constraints. IEEE Trans Neural Netw 19(12):789–798

    MathSciNet  Google Scholar 

  29. 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(5):812–824

    Article  Google Scholar 

  30. Liu Q, Dang C, Huang T (2013) A one-layer recurrent neural network for real-time portfolio optimization with probability criterion. IEEE Trans Cybern 43(1):14–23

    Article  Google Scholar 

  31. Liu Q, Huang T, Wang J (2017) One-layer continuous-and discrete-time projection neural networks for solving variational inequalities and related optimization problems. IEEE Trans Neural Netw Learn Syst 25(7):1308–1318

    Article  Google Scholar 

  32. Guo Z, Wang J (2010) A neurodynamic optimization approach to constrained sparsity maximization based on alternative objective functions. In: The 2010 International joint conference on neural networks (IJCNN)

  33. Ha NT, Strodiot JJ (2018) On the global exponential stability of a projected dynamical system for strongly pseudomonotone variational inequalities. Optim Lett

  34. Bazraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming theory and algorithms. John Wiley, New York

    Google Scholar 

  35. Kennedy MP, Chua LO (1998) Neural networks for nonlinear programming. IEEE Trans Circuits Syst 35(5):554–562

    Article  MathSciNet  Google Scholar 

  36. Xia Y (1996) A new neural network for solving linear and quadratic programming problems. IEEE Trans Neural Netw 7(6):1544–1548

    Article  Google Scholar 

  37. Long C, Hou ZG, Min T (2007) A recurrent neural network for non-smooth nonlinear programming problems. In: International joint conference on neural networks

  38. Pattananupong U, Chaiyaratana N, Tongpadungrod R (2007) Genetic programming and neural networks as interpreters for a distributive tactile sensing system. In: IEEE congress on evolutionary computation

  39. Wu HC (2010) Duality theory for optimization problems with interval-valued objective functions. J Optim Theory Appl 144(3):615–628

    Article  MathSciNet  Google Scholar 

  40. Ishibuchi H, Tanaka H (1990) Multiobjective programming in optimization of the interval objective function. Eur J Oper Res 48(2):219–225

    Article  Google Scholar 

  41. Kinderlehrer D, Stampacchia G (2012) An introduction to variational inequalities and their applications. Academic

  42. Xu D, Xia Y, Mandic DP (2016) Optimization in quaternion dynamic systems: gradient, hessian, and learning algorithms. IEEE Trans Neural Netw Learn Syst Sons 27(2):249–261

    Article  MathSciNet  Google Scholar 

  43. Ghadimi S, Lan G, Zhang H (2019) Generalized uniformly optimal methods for nonlinear programming. J Sci Comput 79:1854–1881

    Article  MathSciNet  Google Scholar 

  44. Terletskyi D, Provotar A (2015) Object-oriented dynamic networks. Cybern Syst Anal 51(1):34–40

    Article  MathSciNet  Google Scholar 

  45. Fukushima M (1992) Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math Program 53(1–3):99–110

    Article  MathSciNet  Google Scholar 

  46. Pang JS (2017) A posteriori error bounds for the linearly-constrained variational inequality problem. Math Oper Res 12(3):474–484

    Article  MathSciNet  Google Scholar 

  47. Qin S, Xue X (2015) A two-layer recurrent neural network for nonsmooth convex optimization problems. IEEE Trans Neural Netw Learn Syst 26(6):1149

    Article  MathSciNet  Google Scholar 

  48. Boese FG (2015) On the asymptotical stability of multivariate dynamical systems. PAMM 1(1):107–108

    Article  Google Scholar 

  49. Bertsekas, D, Tsitsiklis J (2010) Parallel and distributed computation: numerical methods. Computer Science

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Acknowledgements

The work described in this paper was supported by the National Natural Science Foundation of China under Grants 61773004, the Natural Science Foundation of Chongqing Municipality of China under Grant cstc2019jcyj-msxmX0722, Technology Research Foundation of Chongqing Educational Committee under Grants KJQN201900530, Team Building Project for Graduate Tutors in Chongqing under Grants JDDSTD201802, the Postgraduate Research and Innovation Project of Chongqing under Grants 2020ST001, Group Building Scientific Innovation Project for universities in Chongqing CXQT21021 and the Venture & Innovation Support Program for Chongqing Overseas Returnees under Grant cx201803, cx2019155, cx2019127.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Yueqiu Li, Bing Li & Jin Hu

  2. College of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Chunna Zeng

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  1. Yueqiu Li
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  2. Chunna Zeng
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  3. Bing Li
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  4. Jin Hu
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Correspondence to Jin Hu.

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Li, Y., Zeng, C., Li, B. et al. A One-Layer Recurrent Neural Network for Interval-Valued Optimization Problem with Linear Constraints. Neural Process Lett 54, 1275–1292 (2022). https://doi.org/10.1007/s11063-021-10681-w

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