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A Capable Neural Network Framework for Solving Degenerate Quadratic Optimization Problems with an Application in Image Fusion

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Abstract

This paper presents a dynamic optimization scheme for solving degenerate convex quadratic programming (DCQP) problems. According to the saddle point theorem, optimization theory, convex analysis theory, Lyapunov stability theory and LaSalle invariance principle, a neural network model based on a dynamic system model is constructed. The equilibrium point of the model is proved to be equivalent to the optimal solution of the DCQP problem. It is also shown that the network model is stable in the Lyapunov sense and it is globally convergent to an exact optimal solution of the original problem. Several practical examples are provided to show the feasibility and the efficiency of the method.

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References

  1. Agrawal SK, Fabien BC (1999) Optimization of dynamic systems. Kluwer Academic Publishers, Dordrecht

    Book  MATH  Google Scholar 

  2. Avriel M (1976) Nonlinear programming: analysis and methods. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  3. Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming-theory and algorithms, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  4. Bertsekas DP (1989) Parallel and distributed computation: numerical methods. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  5. Cornuejols G, Tutuncu R (2006) Optimization methods in finance. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  6. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  7. Fletcher R (1981) Practical methods of optimization. Wiley, New York

    MATH  Google Scholar 

  8. Zhang H, Liu D, Luo Y, Wang D (2013) Adaptive dynamic programming for control: algorithms and stability. Springer, London

    Book  MATH  Google Scholar 

  9. Yoshikawa T (1990) Foundations of robotics: analysis and control. MIT Press, Cambridge

    Google Scholar 

  10. Kalouptisidis N (1997) Signal processing systems, theory and design. Wiley, New York

    Google Scholar 

  11. Harker PT, Pang JS (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms, and applications. Math Program Ser B 48:161–220

    Article  MathSciNet  MATH  Google Scholar 

  12. He BS, Liao L-Z (2002) Improvements of some projection methods for monotone nonlinear variational inequalities. J Optim Theory Appl 112:111–128

    Article  MathSciNet  MATH  Google Scholar 

  13. Solodov MV, Tseng P (1996) Modified projection-type methods for monotone variational inequalities. SIAM J Control Optim 34:1814–1830

    Article  MathSciNet  MATH  Google Scholar 

  14. Tseng P (2000) A modified forward–backward splitting method for maximal monotone mappings. SIAM J Control Optim 38:431–446

    Article  MathSciNet  MATH  Google Scholar 

  15. Cichocki A, Unbehauer R (1993) Neural networks for optimization with bounded constraints. IEEE Trans Neural Netw 4:293–304

    Article  Google Scholar 

  16. Tank DW, Hopfield JJ (1986) Simple neural optimization networks: an A/D converter, signal decision circuit, and a linear programming pircuit. IEEE Trans Circuits Syst 33:533–541

    Article  Google Scholar 

  17. Kennedy MP, Chua LO (1988) Neural networks for nonlinear programming. IEEE Trans Circuits Syst 35:554–562

    Article  MathSciNet  Google Scholar 

  18. Ding K, Huang N-J (2008) A new class of interval projection neural networks for solving interval quadratic program. Chaos Soliton Fract 35:718–725

    Article  MATH  Google Scholar 

  19. Effati S, Nazemi AR (2006) Neural network models and its application for solving linear and quadratic programming problems. Appl Math Comput 172:305–331

    MathSciNet  MATH  Google Scholar 

  20. Effati S, Ghomashi A, Nazemi AR (2007) Application of projection neural network in solving convex programming problems. Appl Math Comput 188:1103–1114

    MathSciNet  MATH  Google Scholar 

  21. Forti M, Nistri P, Quincampoix M (2006) Convergence of neural networks for programming problems via a nonsmooth Lojasiewicz inequality. IEEE Trans Neural Netw 17:1471–1486

    Article  Google Scholar 

  22. Friesz TL, Bernstein DH, Mehta NJ, Tobin RL, Ganjlizadeh S (1994) Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper Res 42:1120–1136

    Article  MathSciNet  MATH  Google Scholar 

  23. Gao XB, Liao L-Z, Qi LQ (2005) A novel neural network for variational inequalities with linear and nonlinear constraints. IEEE Trans Neural Netw 16:1305–1317

    Article  Google Scholar 

  24. Hu X (2009) Applications of the general projection neural network in solving extended linear-quadratic programming problems with linear constraints. Neurocomputing 72:1131–1137

    Article  Google Scholar 

  25. 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 37:1414–1421

    Article  Google Scholar 

  26. Huang B, Zhang H, Gong D, Wang Z (2013) A new result for projection neural networks to solve linear variational inequalities and related optimization problems. Neural Comput Appl 23:357–362

    Article  Google Scholar 

  27. Liao L, Qi H, Qi L (2001) Solving nonlinear complementarity problems with neural networks: a reformulation method approach. J Comput Appl Math 131:343–359

    Article  MathSciNet  MATH  Google Scholar 

  28. Lillo WE, Loh MH, Hui S, Zăk SH (1993) On solving constrained optimization problems with neural networks: a penalty method approach. IEEE Trans Neural Netw 4:931–939

    Article  Google Scholar 

  29. Liu QS, Wang J (2008) A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming. IEEE Trans Neural Netw 19:558–570

    Article  Google Scholar 

  30. Maa CY, Shanblatt MA (1992) Linear and quadratic programming neural network analysis. IEEE Trans Neural Netw 3:580–594

    Article  Google Scholar 

  31. Malek A, Hosseinipour-Mahani N, Ezazipour S (2010) Efficient recurrent neural network model for the solution of general nonlinear optimization problems. Optim Methods Softw 25:1–18

    Article  MathSciNet  MATH  Google Scholar 

  32. Nazemi AR (2012) A dynamic system model for solving convex nonlinear optimization problems. Commun Nonlinear Sci Numer Simul 17:1696–1705

    Article  MathSciNet  MATH  Google Scholar 

  33. Nazemi AR (2014) A neural network model for solving convex quadratic programming problems with some applications. Eng Appl Artif Intell 32:54–62

    Article  Google Scholar 

  34. Nazemi AR (2011) A dynamical model for solving degenerate quadratic minimax problems with constraints. J Comput Appl Math 236:1282–1295

    Article  MathSciNet  MATH  Google Scholar 

  35. Nazemi AR, Omidi F (2012) A capable neural network model for solving the maximum flow problem. J Comput Appl Math 236:3498–3513

    Article  MathSciNet  MATH  Google Scholar 

  36. Nazemi AR, Omidi F (2013) An efficient dynamic model for solving the shortest path problem. Transp Res Part C Emerg Technol 26:1–19

    Article  Google Scholar 

  37. Nazemi AR, Sharifi E (2013) Solving a class of geometric programming problems by an efficient dynamic model. Commun Nonlinear Sci Numer Simul 18:692–709

    Article  MathSciNet  MATH  Google Scholar 

  38. Tao Q, Cao J, Sun D (2001) A simple and high performance neural network for quadratic programming problems. Appl Math Comput 124:251–260

    MathSciNet  MATH  Google Scholar 

  39. Wu H, Shi R, Qin L, Tao F, He L (2010) A nonlinear projection neural network for solving interval quadratic programming problems and its stability analysis. Math Probl Eng 2010:1–13

    MathSciNet  MATH  Google Scholar 

  40. Xia YS (1996) A new neural network for solving linear and quadratic programming problems. IEEE Trans Neural Netw 7:1544–1547

    Article  Google Scholar 

  41. Xia Y, Wang J, Fok L-M (2004) Grasping-force optimization for multifingered robotic hands using a recurrent neural network. IEEE Trans Robot Autom 20:549–554

    Article  Google Scholar 

  42. Xia Y (1996) A new neural network for solving linear and quadratic programming problems. IEEE Trans Neural Netw 7:1544–1547

    Article  Google Scholar 

  43. Xia Y, Wang J (2004) A general projection neural network for solving monotone variational inequality and related optimization problems. IEEE Trans Neural Netw 15:318–328

    Article  Google Scholar 

  44. Xia YS, Wang J (2000) A recurrent neural network for solving linear projection equations. Neural Netw 13:337–350

    Article  Google Scholar 

  45. Xia Y, Wang J (2004) A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints. IEEE Trans Circuits Syst 51:447–458

    MathSciNet  MATH  Google Scholar 

  46. Xia Y, Wang J (2000) On the stability of globally projected dynamical systems. J Optim Theory Appl 106:129–150

    Article  MathSciNet  MATH  Google Scholar 

  47. Xia Y, Feng G, Wang J (2004) A recurrent neural network with exponential convergence for solving convex quadratic program and related linear piecewise equation. Neural Netw 17:1003–1015

    Article  MATH  Google Scholar 

  48. Xia Y, Feng G (2005) An improved network for convex quadratic optimization with application to real-time beamforming. Neurocomputing 64:359–374

    Article  Google Scholar 

  49. Xue X, Bian W (2007) A project neural network for solving degenerate convex quadratic program. Neurocomputing 70:2449–2459

    Article  Google Scholar 

  50. Xue X, Bian W (2009) A project neural network for solving degenerate quadratic minimax problem with linear constraints. Neurocomputing 72:1826–1838

    Article  Google Scholar 

  51. Yang Y, Cao J (2006) Solving quadratic programming problems by delayed projection neural network. IEEE Trans Neural Netw 17:1630–1634

    Article  Google Scholar 

  52. Yang Y, Cao J (2006) A delayed neural network method for solving convex optimization problems. Int J Neural Syst 16:295–303

    Article  Google Scholar 

  53. Yang Y, Cao J (2008) A feedback neural network for solving convex constraint optimization problems. Appl Math Comput 201:340–350

    MathSciNet  MATH  Google Scholar 

  54. Zhang H, Huang B, Gong D, Wang Z (2013) New results for neutral-type delayed projection neural network to solve linear variational inequalities. Neural Comput Appl 23:1753–1761

    Article  Google Scholar 

  55. Zhang Z, Li C, He X, Huang T (2017) A discrete-time projection neural network for solving degenerate convex quadratic optimization. Circuits Syst Signal Process 36:389–403

    Article  MATH  Google Scholar 

  56. Yang Y, Cao J, Xu X, Hu M, Gao Y (2014) A new neural network for solving quadratic programming problems with equality and inequality constraints. Math Comput Simul 101:103–112

    Article  MathSciNet  Google Scholar 

  57. Sha C, Zhao H, Ren F (2015) A new delayed projection neural network for solving quadratic programming problems with equality and inequality constraints. Neurocomputing 168:1164–1172

    Article  Google Scholar 

  58. Quarteroni A, Sacco R, Saleri F (2007) Numerical mathematics. Texts in applied mathematics, vol. 37, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  59. Miller RK, Michel AN (1982) Ordinary differential equations. Academic Press, NewYork

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  61. Hale JK (1969) Ordinary differential equations. Wiley, New York

    MATH  Google Scholar 

  62. Gass SI, Vinjamuri S (2004) Cycling in linear programming problems. Comput Oper Res 31:303–311

    Article  MathSciNet  MATH  Google Scholar 

  63. Xia Y, Leung H, Boss E (2002) Neural data fusion algorithms based on a linearly constrained least square method. IEEE Trans Neural Netw 13:320–329

    Article  Google Scholar 

  64. Zhang D, Yu L (2012) Exponential state estimation for Markovian jumping neural networks with time-varying discrete and distributed delays. Neural Netw 35:103–111

    Article  MathSciNet  MATH  Google Scholar 

  65. Zhang D, Yu L, Wang Q-G, Ong C-J (2012) Estimator design for discrete-time switched neural networks with asynchronous switching and time-varying delay. IEEE Trans Neural Netw Learn Syst 23:827–834

    Article  Google Scholar 

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Acknowledgements

This work was supported by a research grant (ID: 23081) of Shahrood University of Technology.

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Authors and Affiliations

  1. Department of Mathematics, School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran

    Alireza Nazemi

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  1. Alireza Nazemi
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Correspondence to Alireza Nazemi.

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Nazemi, A. A Capable Neural Network Framework for Solving Degenerate Quadratic Optimization Problems with an Application in Image Fusion. Neural Process Lett 47, 167–192 (2018). https://doi.org/10.1007/s11063-017-9640-4

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  • DOI: https://doi.org/10.1007/s11063-017-9640-4

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