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Flattened aggregate function method for nonlinear programming with many complicated constraints

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In this paper, efforts are made to solve nonlinear programming with many complicated constraints more efficiently. The constrained optimization problem is firstly converted to a minimax problem, where the max-value function is approximately smoothed by the so-called flattened aggregate function or its modified version. For carefully updated aggregate parameters, the smooth unconstrained optimization problem is solved by an inexact Newton method. Because the flattened aggregate function can usually reduce greatly the amount of computation for gradients and Hessians, the method is more efficient. Convergence of the proposed method is proven and some numerical results are given to show its efficiency.

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Fig. 1


  1. 1.

    Bagirov, A.M., Al Nuaimat, A., Sultanova, N.: Hyperbolic smoothing function method for minimax problems. Optimization 62, 759–782 (2013)

  2. 2.

    Bandler, J.W., Charalambous, C.: Nonlinear programming using minimax techniques. J. Optim. Theory Appl. 13, 607–619 (1974)

  3. 3.

    Bertsekas, D.P.: Approximation procedures based on the method of multipliers. J. Optim. Theory Appl. 23, 487–510 (1977)

  4. 4.

    Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)

  5. 5.

    Boggs, P.T., Tolle, J.W.: Sequential quadratic programming. Acta Numer. 4, 1–51 (1995)

  6. 6.

    Dembo, R.S., Steihaug, T.: Truncated-newton algorithms for large-scale unconstrained optimization. Math. Program. 26, 190–212 (1983)

  7. 7.

    Forsgren, A., Gill, P.E., Wright, M.H.: Interior methods for nonlinear optimization. SIAM Rev. 44, 525–597 (2003)

  8. 8.

    Gigola, C., Gomez, S.: A regularization method for solving the finite convex min-max problem. SIAM J. Numer. Anal. 27, 1621–1634 (1990)

  9. 9.

    Gould, N., Orban, D., Toint, P.: Numerical methods for large-scale nonlinear optimization. Acta Numer. 14, 299–361 (2005)

  10. 10.

    Han, S.P., Mangasarian, O.L.: Exact penalty functions in nonlinear programming. Math. Program. 17, 251–269 (1979)

  11. 11.

    Herskovits, J.: A two-stage feasible directions algorithm for nonlinear constrained optimization. Math. Program. 36, 19–38 (1986)

  12. 12.

    Jian, J.B.: Fast Algorithm for Smoothing Constrained Optimization: Theoretical Analysis and Numerical Experiments. Science Press, Beijing (2010)

  13. 13.

    Kort, B.W., Bertsakas, D.P.: A new penalty function algorithm algorithm for constrained minimization. In: Proceedings of the 1972 IEEE Conference on Decision and Control, New Orleans (1972)

  14. 14.

    Li, D.H., Qi, L.Q., Tam, J., Wu, S.Y.: A smoothing Newton method for semi-infinite programming. J. Global Optim 30, 169–194 (2004)

  15. 15.

    Li, J.X., Huo, J.Z.: Inexact smoothing method for large scale minimax optimization. Appl. Math. Comput. 218, 2750–2760 (2011)

  16. 16.

    Li, X.S.: An aggregate function method for nonlinear programming. Sci. China (A) 34, 1467–1473 (1991)

  17. 17.

    Li, X.S., Fang, S.C.: On the entropic regularization method for solving min-max problems with applications. Math. Methods Oper. Res. 46, 119–130 (1997)

  18. 18.

    Mayne, D.Q., Polak, E.: Feasible direction algorithm for optimization problems with equality and inequality constraints. Math. Program. 11, 67–80 (1976)

  19. 19.

    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, New York (2006)

  20. 20.

    Pee, E.Y., Royset, J.O.: On solving large-scale finite minimax problems using exponential smoothing. J. Optim. Theory Appl. 148, 390–421 (2011)

  21. 21.

    Peng, J.M., Lin, Z.: A non-interior continuation method for generalized linear complementarity problems. Math. Program. 86, 533–563 (1999)

  22. 22.

    Polak, E., Royset, J.O., Womersley, R.S.: Algorithms with adaptive smoothing for finite minimax problems. J. Optim. Theory Appl. 119, 459–484 (2003)

  23. 23.

    Polak, E., Womersley, R.S., Yin, H.X.: An algorithm based on active sets and smoothing for discretized semi-infinite minimax problems. J. Optim. Theory Appl. 138, 311–328 (2008)

  24. 24.

    Rustem, B.: Equality and inequality constrained optimization algorithms with convergent stepsizes. J. Optim. Theory Appl. 76, 429–453 (1993)

  25. 25.

    Tang, H.W., Zhang, L.W.: A maximum entropy method for convex programming. Chin. Sci. Bull. 40, 361–364 (1995)

  26. 26.

    Wang, Y.C., Tang, H.W.: Investigation of maximum entropy method for min-max problems (I). J. Dalian Univ. Technol. 37, 495–499 (1997)

  27. 27.

    Wang, Y.C., Tang, H.W.: Investigation of maximum entropy method for min-max problems (II). J. Dalian Univ. Technol. 38, 1–5 (1998)

  28. 28.

    Xiao, Y., Yu, B.: A truncated aggregate smoothing newton method for minimax problems. Appl. Math. Comput. 216, 1868–1879 (2010)

  29. 29.

    Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20, 267–279 (2001)

  30. 30.

    Yu, B., Feng, G.C., Zhang, S.L.: The aggregate constraint homotopy method for nonconvex nonlinear programming. Nonlinear Anal. 45, 839–847 (2001)

  31. 31.

    Yuan, Y.X., Sun, W.Y.: Optimization Theory and Methods. Science Press, Beijing (1999)

  32. 32.

    Zang, I.: A smoothing technique for min-max optimization. Math. Program. 19, 61–77 (1980)

  33. 33.

    Zhang, L.L., Zhang, P.A., Li, X.S.: Some notes on the entropic method. Oper. Res. Manag. Sci. 18, 74–77 (2009)

  34. 34.

    Zhao, G., Wang, Z., Mou, H.: Uniform approximation of min/max functions by smooth splines. J. Comput. Appl. Math. 236, 699–703 (2011)

  35. 35.

    Zhou, Z.Y.: Smoothing homotopy methods for solving several mathematieal programming problems. Dalian University of Technology, Ph.D, diss. (2011)

  36. 36.

    Zhou, Z.Y., Yu, B.: The flattened aggregate constraint homotopy method for nonlinear programming problems with many nonlinear constraints. Abstr. Appl. Anal. 4, 1–14 (2014)

  37. 37.

    Dolan, E., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

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This work is financially supported by the Innovation Talent Training Program of Science and Technology of Jilin Province of China (20180519011JH), the Science and Technology Development Project Program of Jilin Province (20190303132SF), and the Project of Education Department of Jilin Province (JJKH20200028KJ).

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Correspondence to Yueting Yang.

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Jiang, X., Yang, Y., Lu, Y. et al. Flattened aggregate function method for nonlinear programming with many complicated constraints. Numer Algor (2020). https://doi.org/10.1007/s11075-020-00881-1

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  • Nonlinear programming
  • Complicated constraints
  • Minimax problem
  • Flattened aggregate function
  • Inexact Newton method

Mathematics Subject Classification (2010)

  • 90C30
  • 65K05