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A new collaborate neuro-dynamic framework for solving convex second order cone programming problems with an application in multi-fingered robotic hands

  • Alireza NazemiEmail author
Article
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Abstract

A neural network model is constructed on the basis of the duality theory, optimization theory, convex analysis theory and Lyapunov stability theory to solve convex second-order cone programming (CSOCP) problems. According to Karush-Kuhn-Tucker conditions of convex optimization, the equilibrium point of the proposed neural network is proved to be equivalent to the optimal solution of the CSOCP problem. By employing Lyapunov function approach, it is also shown that the presented neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the original optimization problem. Simulation results show that the neural network is feasible and efficient.

Keywords

Neural network Second-order cone programming Convex programming Convergence Stability 

Notes

Funding

This study was not funded by any grant.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.

References

  1. 1.
    Faraut J, Kornyi A (1994) Analysis on symmetric cones. In: Oxford mathematical monographs. Oxford University Press, New YorkGoogle Scholar
  2. 2.
    Alizadeh F, Goldfarb D (2003) Second-order cone programming. Math Program 95:3–51MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Application of second-order cone programming. Linear Algebra Appl 284:193–228MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andersen ED, Roos C, Terlaky T (2003) On implementing a primal–dual interior-point method for conic quadratic optimization. Math Program 95:249–277MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Monteiro RDC, Tsuchiya T (2000) Polynomial convergence of primal–dual algorithms for the second-order cone programs based on the MZ-family of directions. Math Program 88:61–83MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Tsuchiya T (1999) A convergence analysis of the scaling-invariant primal–dual path-following algorithms for second-order cone programming. Optim Methods Softw 11:141–182MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen X-D, Sun D, Sun J (2003) Complementarity functions and numerical experiments for second-order cone complementarity problems. Comput Optim Appl 25:39–56MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fukushima M, Luo Z-Q, Tseng P (2002) Smoothing functions for second-order cone complementarity problems. SIAM J Optim 12:436–460MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hayashi S, Yamashita N, Fukushima M (2005) A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J Optim 15:593– 615MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kanzow C, Ferenczi I, Fukushima M (2009) On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J Optim 20:297–320MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen J. -S., Tseng P (2005) An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math Program 104:293–327MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pana S, Chenb J-S (2010) Interior proximal methods and central paths for convex second-order cone programming. Nonlinear Anal 73:3083–3100MathSciNetCrossRefGoogle Scholar
  13. 13.
    Boyd SP, Wegbreit B (2007) A fast computation of optimal contact forces. IEEE Trans Robot 23 (6):1117–1132CrossRefGoogle Scholar
  14. 14.
    Boyd S, Crusius C, Hansson A (1998) Control applications of nonlinear convex programming. J Control Process 8(5):313–324CrossRefGoogle Scholar
  15. 15.
    Bertsimas D, Brown DB (2007) Constrained stochastic LQC: atractableapproach. IEEE Trans Autom Control 52(10):1826–1841CrossRefzbMATHGoogle Scholar
  16. 16.
    Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear programming- theory and algorithms, 2nd. Wiley, New YorkzbMATHGoogle Scholar
  17. 17.
    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 33:533–541CrossRefGoogle Scholar
  18. 18.
    Kennedy MP, Chua LO (1988) Neural networks for nonlinear programming. IEEE Trans Circuits Syst 35:554–562MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chen KZ, Leung Y, Leung KS, Gao XB (2002) A neural network for nonlinear programming problems. Neural Comput Applic 11:103–111CrossRefGoogle Scholar
  20. 20.
    Effati S, Nazemi AR (2006) Neural network models and its application for solving linear and quadratic programming problems. Appl Math Comput 172:305–331MathSciNetzbMATHGoogle Scholar
  21. 21.
    Effati S, Ghomashi A, Nazemi AR (2007) Application of projection neural network in solving convex programming problems. Appl Math Comput 188:1103–1114MathSciNetzbMATHGoogle Scholar
  22. 22.
    Forti M, Nistri P, Quincampoix M (2004) Generalized neural network for nonsmooth nonlinear programming problems. IEEE Trans Circuits Syst I 51:1741–1754MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    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 4:931–939CrossRefGoogle Scholar
  24. 24.
    Malek A, Hosseinipour-Mahani N, Ezazipour S (2010) Efficient recurrent neural network model for the solution of general nonlinear optimization problems. Optim Methods Softw 25:1–18MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nazemi AR (2012) A dynamic system model for solving convex nonlinear optimization problems. Commun Nonlinear Sci Numer Simul 17:1696–1705MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nazemi AR (2013) Solving general convex nonlinear optimization problems by an efficient neurodynamic model. Eng Appl Artif Intel 26:685–696CrossRefGoogle Scholar
  27. 27.
    Nazemi AR, Effati S (2013) An application of a merit function for solving convex programming problems. Comput Ind Eng 66:212–221CrossRefGoogle Scholar
  28. 28.
    Nazemi AR, Nazemi M (2014) A gradient-based neural network method for solving strictly convex quadratic programming problems. Cogn Comput 6(3):484–495MathSciNetCrossRefGoogle Scholar
  29. 29.
    Xia Y, Wang J (2005) A recurrent neural network for solving nonlinear convex programs subject to linear constraints. IEEE Trans Neural Netw 16:379–386CrossRefGoogle Scholar
  30. 30.
    Xue X, Bian W (2007) A project neural network for solving degenerate convex quadratic program. Neural Netw 70:2449–2459Google Scholar
  31. 31.
    Xue X, Bian W (2008) Subgradient-based neural network for nonsmooth convex optimization problems. IEEE Trans Circuits Syst I 55:2378–2391MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yanga Y, Cao J (2010) The optimization technique for solving a class of non-differentiable programming based on neural network method. Nonlinear Anal 11:1108–1114MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yang Y, Xu X (2007) The projection neural network for solving convex nonlinear programming. In: Huang D-S, Heutte L, Loog M (eds) ICIC 2007, LNAI 4682. Springer, Berlin, pp 174–181Google Scholar
  34. 34.
    Yang Y, Cao J (2008) A feedback neural network for solving convex constraint optimization problems. Appl Math Comput 201:340–350MathSciNetzbMATHGoogle Scholar
  35. 35.
    Mu X, Liu S, Zhang Y (2005) A neural network algorithm for second-order conic programming. In: Proceedings of the 2nd international symposiumon neural networks, Chongqing, China, Part II, pp 718–724Google Scholar
  36. 36.
    Xia Y, Wang J, Fok LM (2004) Grasping-force optimization formultifingered robotic hands using a recurrent neural network. IEEE Trans Robot Autom 20(3):549–554CrossRefGoogle Scholar
  37. 37.
    Ko C-H, Chen J-S, Yang C-Y (2011) Recurrent neural networks for solving second-order cone programs. Neurocomputing 74:3646–3653CrossRefGoogle Scholar
  38. 38.
    Sun J, Zhang L (2009) A globally convergent method based on Fischer–Burmeister operators for solving second-order cone constrained variational inequality problems. Comput Math Appl 58:1936–1946MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sun J, Chen J-S, Ko C-H (2012) Neural networks for solving second-order cone constrained variational inequality problem. Comput Optim Appl 51:623–648MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Miao X, Chen J-S, Ko C-H (2014) A smoothed NR neural network for solving nonlinear convex programs with second-order cone constraints. Inf Sci 268:255–270MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Miao X, Chen J-S, Ko C-H (2016) Neural network based on the generalized FB function for nonlinear convex programs with second-order cone constraints. Neurocomputing 203:62–72CrossRefGoogle Scholar
  42. 42.
    Zhang Y (2017) A projected-based neural network method for second-order cone programming. International Journal of Machine Learning and Cybernetics 8(6):1907–1914CrossRefGoogle Scholar
  43. 43.
    Liao LZ, Qi HD (1999) A neural network for the linear complementarity problem. Math Comput Model 29(3):9–18MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Benson H. Y., Vanderbei RJ (2003) Solving problems with semidefinite and related constraints using interior-point methods for nonlinear programming. Math Program Ser B 95:279–302MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Miller RK, Michel AN (1982) Ordinary differential equations. Academic Press, NewYorkzbMATHGoogle Scholar
  46. 46.
    Yang Y, Cao J, Xua X, Hua M, Gao Y (2014) A new neural network for solving quadratic programming problemswith equality and inequality constraints. Math Comput Simul 101:103–112CrossRefGoogle Scholar
  47. 47.
    Nazemi AR (2017) A capable neural network framework for solving degenerate quadratic optimization problems with an application in image fusion. Neural Processing Letters 47(1):167–192CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesShahrood University of TechnologyShahroodIran

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