Applied Intelligence

, Volume 49, Issue 2, pp 396–419 | Cite as

A new gradient-based neural dynamic framework for solving constrained min-max optimization problems with an application in portfolio selection models

  • Alireza Nazemi
  • Marziyeh MortezaeeEmail author


A neural network model based on a nonlinear dynamic model for solving a class of min-max problems, which is motivated as a non-differentiable optimization problem, is proposed. The main idea is to convert the non-differentiable problem into an equivalent differentiable convex optimization problem using a smoothing scheme called an entropy procedure. A neural network model is then constructed for solving the obtained convex problem. The stability of the equilibrium point and the convergence of the optimal solution are discussed. As an application in economics, we use the proposed scheme to a min-max portfolio optimization problem. Several clarifying examples and simulation results are provided to demonstrate the correctness of the results and the good performance of the presented model.


Min-Max optimization Neural network Entropy function Dynamic model Convex programming Convergent Stability Portfolio selection 


Compliance with Ethical Standards

Conflict of interests

Disclosure of potential conflicts of interest. The authors declare that they have no conflict of interest.

Informed Consent

Informed consent was obtained from all individual participants included in the study.

Consent for Publication

Chapters don’t contain any studies with human participants performed by any of the authors. Chapters don’t contain any studies with animals performed by any of the authors. Chapters don’t contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Agnew D (1981) Improved minimax optimization for circuit design. IEEE Transactions on Circuits and Systems 28:791–803MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chao M, Wang Z, Liang Y, Hu Q (2008) Quadratically constraint quadratical algorithm model for nonlinear minimax problems. Appl Math Comput 205:247–262MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cherkaev E, Cherkaev A (2008) Minimax optimization problem of structural design. Comput Struct 86:1426–1435CrossRefzbMATHGoogle Scholar
  4. 4.
    Teo KL, Yang XQ (2001) Portfolio selection problem with minimax type risk function. J Ann Oper Res 101:333–349MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Di Pillo G, Grippo L, Lucidi S (1997) Smooth transformation of the generalized minimax problem. J Optim Theory Appl 95:1– 24MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cornuejols G, Tutuncu R (2006) Optimization methods in finance. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  7. 7.
    Charalambous C, Conn AR (1978) An efficient method to solve the minimax problem directly. SIAM J Numer Anal 15:162–187MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gigola C, Gomez S (1990) A regularization method for solving the finite convex min-max problem. SIAM J Numer Anal 27:1621–1634MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jian J, Tang C (2005) An SQP feasible desent algorithm for nonlinear inequality constrained optimization without strict complementarity. Comput Math Appl 49:223–238MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jian J, Quan R, Zhang X (2007) Feasible generalized monotone line search SQP algorithm for nonlinear minimax problems with inequality constraints. Comput Appl Math 205:406–429MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Polak GG, Rogers DF, Sweeney DJ (2010) Risk management strategies via minimax portfolio optimization. Eur J Oper Res 207:409–419MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li X (1994) An efficient approach to a class of non-smooth optimization problems. Science in China (Series A) 37:323–330MathSciNetzbMATHGoogle Scholar
  13. 13.
    Liang JJ, Suganthan PN, Deb K (2005) Novel composition test functions for numerical global optimization. In: Proceedings 2005 IEEE on swarm intelligence symposium, 2005. SIS 2005. IEEEGoogle Scholar
  14. 14.
    Srinivas N, Deb K (1994) Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2.3:221–248CrossRefGoogle Scholar
  15. 15.
    Deb K, Padhye N (2014) Enhancing performance of particle swarm optimization through an algorithmic link with genetic algorithms. Comput Optim Appl 57:761–794MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Padhye N, Bhardawaj P, Deb K (2013) Improving differential evolution through a unified approach. J Glob Optim 55:771–799MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li X, Fang S (1997) On the entropy regularization method for solving min-max problems with applications. Math Methods Oper Res 46:119–130MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Madsen K, Jacobsen HS (1978) Linear constrained minimax optimization. Math Program 14:208–223CrossRefzbMATHGoogle Scholar
  19. 19.
    Rustem B, Nguyen Q (1998) An algorithm for the inequality-constrained discrete minimax problem. SIAM J Optim 8:265–283MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shen Z, Huang Z, Wolfe M (1997) An interval maximum entropy method for a discrete minimax problem. Appl Math Comput 87:49–68MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vardi A (1992) New minimax algorithm. J Optim Theory Appl 75:613–634MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Xingsia L (1992) An entropy-based aggregate method for minimax optimization. Eng Optim 18:277–285CrossRefGoogle Scholar
  23. 23.
    Xue W, Shen C, Pu D (2009) A new non-monotone SQP algorithm for the minimax problem. Int J Comput Math 86:1149–1159MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ye F, Liu H, Zhou S, Liu S (2008) A smoothing trust-region newton-CG method for minimax problem. Appl Math Comput 199:581–589MathSciNetzbMATHGoogle Scholar
  25. 25.
    Yu YH, Gao L (2002) Nonmonotone linear search algorithm for constrained minimax problem. J Optim Theory Appl 115:419– 446MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhu Z, Cai X, Jian J (2009) An improved SQP algorithm for solving minimax problems. Appl Math Lett 22:464–469MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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 33:533–541CrossRefGoogle Scholar
  28. 28.
    Chen JS, Ko CH, Pan S (2010) A neural network based on the generalized Fischer–Burmeister function for nonlinear complementarity problems. Inform Sci 180:697–711MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    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
  30. 30.
    Effati S, Ghomashi A, Nazemi AR (2007) Application of projection neural network in solving convex programming problems. Appl Math Comput 188:1103–1114MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gao X (2004) A novel neural network for nonlinear convex programming. IEEE Trans Neural Netw 15:613–621CrossRefGoogle Scholar
  32. 32.
    Gao X, Liao L (2009) A new projection-based neural network for constrained variational inequalities. IEEE Trans Neural Netw 20:373–388CrossRefGoogle Scholar
  33. 33.
    Huang YC (2002) A novel method to handle inequality constraints for convex programming neural network. Neural Process Lett 16:17–27CrossRefzbMATHGoogle Scholar
  34. 34.
    Malek A, Hosseinipour-Mahani N, Ezazipour S (2009) Efficient recurrent neural network model for the solution of general nonlinear optimization problems. Optim Methods Softw 25:1–18MathSciNetzbMATHGoogle Scholar
  35. 35.
    Nazemi AR (2011) A dynamical model for solving degenerate quadratic minimax problems with constraints. J Comput Appl Math 236:1282–1295MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Nazemi AR (2012) A dynamic system model for solving convex nonlinear optimization problems. Commun Nonlinear Sci Numer Simul 17:1696–1705MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nazemi AR (2014) A neural network model for solving convex quadratic programming problems with some applications. Eng Appl Artif Intell 32:54–62CrossRefGoogle Scholar
  38. 38.
    Nazemi AR (2013) Solving general convex nonlinear optimization problems by an efficient neurodynamic model. Eng Appl Artif Intell 26:685–696CrossRefGoogle Scholar
  39. 39.
    Nazemi AR, Omidi F (2012) A capable neural network model for solving the maximum flow problem. J Comput Appl Math 236:3498–3513MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nazemi AR, Omidi F (2013) An efficient dynamic model for solving the shortest path problem. Transp Res C 26:1–19CrossRefGoogle Scholar
  41. 41.
    Nazemi AR, Sharifi E (2013) Solving a class of geometric programming problems by an efficient dynamic model. Commun Nonlinear Sci Numer Simul 18:692–709MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nazemi AR, Effati S (2013) An application of a merit function for solving convex programming problems. Comput Ind Eng 66:212–221CrossRefGoogle Scholar
  43. 43.
    Nazemi AR, Tahmasbi N (2013) A high performance neural network model for solving chance constrained optimization problems. Neurocomputing 121:540–550CrossRefGoogle Scholar
  44. 44.
    Xia Y, Wang J (2004) A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints. IEEE Trans Circuits Syst 51:447–458MathSciNetzbMATHGoogle Scholar
  45. 45.
    Xue X, Bian W (2007) A project neural network for solving degenerate convex quadratic program. Neural Netw 70:2449–2459Google Scholar
  46. 46.
    Yang 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
  47. 47.
    Rustem B, Nguyen Q (1998) An algorithm for the inequality-constrained discrete min–max problem. SIAM J Optim 8:265– 283MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Liu Y, Jian J (2015) New active set identification for general constrained optimization and minimax problems. J Math Anal Appl 421:1405–1416MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Yang Y, Cao J, Xu X, Liu J (2012) A generalized neural network for solving a class of minimax optimization problems with linear constraints. Appl Math Comput 218:7528–7537MathSciNetzbMATHGoogle Scholar
  50. 50.
    Bazaraa MS, Sherali HD, Shetty CM (1993) Non-linear programming, theory and algorithms, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  51. 51.
    Hale JK (1969) Ordinary differential equations. Wiley, New YorkzbMATHGoogle Scholar
  52. 52.
    Facchinei F, Pang J (2003) Finite-dimensional variational inequalities and complementarity problems. Springer, New YorkzbMATHGoogle Scholar
  53. 53.
    Sastry S (1999) Nonlinear systems analysis. Stability and Control, SpringerGoogle Scholar
  54. 54.
    Quarteroni A, Sacco R, Saleri F (2007) Numerical mathematics. In: Texts in applied mathematics, 2nd edn., vol 37. Springer, BerlinGoogle Scholar
  55. 55.
    Chen J-S, Gao H-T, Pan S-H (2009) An R-linearly convergent derivative-free algorithm for nonlinear complementarity problems based on the generalized Fischer-Burmeister merit function. J Comput Appl Math 232:455–471MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Chen J-S, Ko C-H, Pan S-H (2010) A neural network based on the generalized Fischer-Burmeister function for nonlinear complementarity problems. Inf Sci 180:697–711MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Chen J-S, Pan S-H (2008) A family of NCP functions and a descent method for the nonlinear comple-mentarity problem. Comput Optim Appl 40:389–404MathSciNetCrossRefGoogle Scholar
  58. 58.
    Sun J, Chen JS, Ko CH (2012) Neural networks for solving second-order cone constrained variational inequality problem. Comput Optim Appl 51:623–648MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Yang Y, Cao J (2008) A feedback neural network for solving convex constraint optimization problems. Appl Math Comput 201:340–350MathSciNetzbMATHGoogle Scholar
  60. 60.
    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
  61. 61.
    Kennedy MP, Chua LO (1988) Neural networks for nonlinear programming. IEEE Trans Circuits Syst 35:554–562MathSciNetCrossRefGoogle Scholar
  62. 62.
    Nazemi AR, Effati S (2013) An application of a merit function for solving convex programming problems. Comput Ind Eng 66:212–221CrossRefGoogle Scholar
  63. 63.
    Cai XQ, Teo KL, Yang XQ, Zhou XY (2000) Portfolio optimization under a minimax rule. Manag Sci 46:957–972CrossRefzbMATHGoogle Scholar
  64. 64.
    Teo KL, Yang XQ (2001) Portfolio selection problem with minimax type risk function. Ann Oper Res 101:333–349MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Yu M, Inoue H, Shi J (2006) Portfolio optimization problems with linear programming models. In: Proceedings of the 2006 China international conference in financeGoogle Scholar
  66. 66.
    Papahristodoulou C, Dotzauer E (2004) Optimal portfolios using linear programming models. J Oper Res Soc 55:1169–1177CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesShahrood University of TechnologyShahroodIran

Personalised recommendations