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**Global convergence of a new nonmonotone filter method for equality constrained optimization.**
*(English)*
Zbl 1266.65105

Summary: A new nonmonotone filter trust region method is introduced for solving optimization problems with equality constraints. This method directly uses the dominated area of the filter as an acceptability criterion for trial points and allows the dominated area decreasing nonmonotonically. Compared with the filter-type method, our method has more flexible criteria and can avoid Maratos effect in a certain degree. Under reasonable assumptions, we prove that the given algorithm is globally convergent to a first order stationary point for all possible choices of the starting point. Numerical tests are presented to show the effectiveness of the proposed algorithm.

### MSC:

65K10 | Numerical optimization and variational techniques |

### Software:

ipfilter
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\textit{K. Su} et al., J. Appl. Math. 2013, Article ID 976509, 9 p. (2013; Zbl 1266.65105)

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### References:

[1] | R. H. Byrd, R. B. Schnabel, and G. A. Shultz, “A trust region algorithm for nonlinearly constrained optimization,” SIAM Journal on Numerical Analysis, vol. 24, no. 5, pp. 1152-1170, 1987. · Zbl 0631.65068 |

[2] | J. E. Dennis Jr., M. El-Alem, and M. C. Maciel, “A global convergence theory for general trust-region-based algorithms for equality constrained optimization,” SIAM Journal on Optimization, vol. 7, no. 1, pp. 177-207, 1997. · Zbl 0867.65031 |

[3] | M. J. D. Powell and Y. Yuan, “A trust region algorithm for equality constrained optimization,” Mathematical Programming A, vol. 49, no. 2, pp. 189-211, 1991. · Zbl 0816.90121 |

[4] | R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty function,” Mathematical Programming A, vol. 91, no. 2, pp. 239-269, 2002. · Zbl 1049.90088 |

[5] | C. M. Chin and R. Fletcher, “On the global convergence of an SLP-filter algorithm that takes EQP steps,” Mathematical Programming A, vol. 96, no. 1, pp. 161-177, 2003. · Zbl 1023.90060 |

[6] | R. Fletcher, N. I. M. Gould, S. Leyffer, P. L. Toint, and A. Wächter, “Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 635-659, 2002. · Zbl 1038.90076 |

[7] | R. Fletcher, S. Leyffer, and P. L. Toint, “On the global convergence of a filter-SQP algorithm,” SIAM Journal on Optimization, vol. 13, no. 1, pp. 44-59, 2002. · Zbl 1029.65063 |

[8] | S. Ulbrich, “On the superlinear local convergence of a filter-SQP method,” Mathematical Programming B, vol. 100, no. 1, pp. 217-245, 2004. · Zbl 1146.90525 |

[9] | M. Ulbrich, S. Ulbrich, and L. N. Vicente, “A globally convergent primal-dual interior-point filter method for nonlinear programming,” Mathematical Programming A, vol. 100, no. 2, pp. 379-410, 2004. · Zbl 1070.90110 |

[10] | R. Fletcher and S. Leyffer, “A bundle filter method for nonsmooth nonlinear optimization,” Tech. Rep. NA/195, Department of Mathematics, University of Dundee, Dundee, Scotland, December 1999. |

[11] | L. Grippo, F. Lampariello, and S. Lucidi, “A nonmonotone line search technique for Newton’s method,” SIAM Journal on Numerical Analysis, vol. 23, no. 4, pp. 707-716, 1986. · Zbl 0616.65067 |

[12] | M. Ulbrich and S. Ulbrich, “Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function,” Mathematical Programming B, vol. 95, no. 1, pp. 103-135, 2003. · Zbl 1030.90123 |

[13] | K. Su and D. Pu, “A nonmonotone filter trust region method for nonlinear constrained optimization,” Journal of Computational and Applied Mathematics, vol. 223, no. 1, pp. 230-239, 2009. · Zbl 1180.65081 |

[14] | K. Su and Z. Yu, “A modified SQP method with nonmonotone technique and its global convergence,” Computers & Mathematics with Applications, vol. 57, no. 2, pp. 240-247, 2009. · Zbl 1165.90684 |

[15] | N. I. M. Gould and P. L. Toint, “Global convergence of a non-monotone trust-region filter algorithm for nonlinear programming,” in Multiscale optimization methods and applications, vol. 82, pp. 125-150, Springer, New York, NY, USA, 2006. · Zbl 1130.90399 |

[16] | Z. Chen and X. Zhang, “A nonmonotone trust-region algorithm with nonmonotone penalty parameters for constrained optimization,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 7-39, 2004. · Zbl 1059.65053 |

[17] | Z. W. Chen, “A penalty-free-type nonmonotone trust-region method for nonlinear constrained optimization,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 1014-1046, 2006. · Zbl 1093.65058 |

[18] | F. A. M. Gomes, M. C. Maciel, and J. M. Martínez, “Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters,” Mathematical Programming A, vol. 84, no. 1, pp. 161-200, 1999. · Zbl 1050.90574 |

[19] | M. J. D. Powell, “Convergence properties of a class of minimization algorithms,” in Nonlinear Programming, pp. 1-27, Academic Press, New York, NY, USA, 1974. · Zbl 0321.90045 |

[20] | K. Schittkowski, More Test Examples for Nonlinear Programming Codes, vol. 282, Springer, Berlin, Germany, 1987. · Zbl 0658.90060 |

[21] | W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, vol. 187, Springer, Berlin, Germany, 1981. · Zbl 0452.90038 |

[22] | M. Lalee, J. Nocedal, and T. Plantenga, “On the implementation of an algorithm for large-scale equality constrained optimization,” SIAM Journal on Optimization, vol. 8, no. 3, pp. 682-706, 1998. · Zbl 0913.65055 |

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