Abstract and Applied Analysis

An Improved Nonmonotone Filter Trust Region Method for Equality Constrained Optimization

Zhong Jin

Full-text: Open access

Abstract

Motivated by the method of Su and Pu (2009), we present an improved nonmonotone filter trust region algorithm for solving nonlinear equality constrained optimization. In our algorithm a modified nonmonotone filter technique is proposed and the restoration phase is not needed. At every iteration, in common with the composite-step SQP methods, the step is viewed as the sum of two distinct components, a quasinormal step and a tangential step. A more relaxed accepted condition for trial step is given and a crucial criterion is weakened. Under some suitable conditions, the global convergence is established. In the end, numerical results show our method is effective.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 163487, 9 pages.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1366306737

Digital Object Identifier
doi:10.1155/2013/163487

Mathematical Reviews number (MathSciNet)
MR3034886

Zentralblatt MATH identifier
1270.90075

Citation

Jin, Zhong. An Improved Nonmonotone Filter Trust Region Method for Equality Constrained Optimization. Abstr. Appl. Anal. 2013 (2013), Article ID 163487, 9 pages. doi:10.1155/2013/163487. https://projecteuclid.org/euclid.aaa/1366306737


Export citation

References

  • R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty function,” Mathematical Programming, vol. 91, no. 2, pp. 239–269, 2002.
  • 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.
  • 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.
  • S. Ulbrich, “On the superlinear local convergence of a filter-SQP method,” Mathematical Programming, vol. 100, no. 1, pp. 217–245, 2004.
  • A. Vardi, “A trust region algorithm for equality constrained minimization: convergence properties and implementation,” SIAM Journal on Numerical Analysis, vol. 22, no. 3, pp. 575–591, 1985.
  • 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.
  • E. O. Omojokun, Trust region algorithms for optimization with nonlinear equality and inequality constraints [Ph.D. thesis], University of Colorado, Boulder, Colo, USA, 1989.
  • C. Audet and J. E. Dennis, “A pattern search filter method for nonlinear programming without derivatives,” SIAM Journal on Optimization, vol. 14, no. 4, pp. 980–1010, 2004.
  • C. M. Chin and R. Fletcher, “On the global convergence of an SLP-filter algorithm that takes EQP steps,” Mathematical Programming, vol. 96, no. 1, pp. 161–177, 2003.
  • M. Ulbrich, S. Ulbrich, and L. N. Vicente, “A globally convergent primal-dual interior-point filter method for nonlinear programming,” Mathematical Programming, vol. 100, no. 2, pp. 379–410, 2004.
  • 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, 1999.
  • E. Karas, A. Ribeiro, C. Sagastizábal, and M. Solodov, “A bundle-filter method for nonsmooth convex constrained optimization,” Mathematical Programming, vol. 116, no. 1-2, pp. 297–320, 2009.
  • N. I. M. Gould, S. Leyffer, and P. L. Toint, “A multidimensional filter algorithm for nonlinear equations and nonlinear least-squares,” SIAM Journal on Optimization, vol. 15, no. 1, pp. 17–38, 2004.
  • N. I. M. Gould, C. Sainvitu, and P. L. Toint, “A filter-trust-region method for unconstrained optimization,” SIAM Journal on Optimization, vol. 16, no. 2, pp. 341–357, 2006.
  • A. Wächter and L. T. Biegler, “Line search filter methods for nonlinear programming: motivation and global convergence,” SIAM Journal on Optimization, vol. 16, no. 1, pp. 1–31, 2005.
  • A. Wächter and L. T. Biegler, “Line search filter methods for nonlinear programming: local convergence,” SIAM Journal on Optimization, vol. 16, no. 1, pp. 32–48, 2005.
  • 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.
  • Z. W. Chen and X. S. 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.
  • N. Y. Deng, Y. Xiao, and F. J. Zhou, “Nonmonotonic trust region algorithm,” Journal of Optimization Theory and Applications, vol. 76, no. 2, pp. 259–285, 1993.
  • P. L. Toint, “Non-monotone trust-region algorithms for nonlinear optimization subject to convex constraints,” Mathematical Programming, vol. 77, no. 3, pp. 69–94, 1997.
  • M. Ulbrich, “Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems,” SIAM Journal on Optimization, vol. 11, no. 4, pp. 889–917, 2001.
  • D. C. Xu, J. Y. Han, and Z. W. Chen, “Nonmonotone trust-region method for nonlinear programming with general constraints and simple bounds,” Journal of Optimization Theory and Applications, vol. 122, no. 1, pp. 185–206, 2004.
  • D. T. Zhu, “A nonmonotonic trust region technique for nonlinear constrained optimization,” Journal of Computational Mathematics, vol. 13, no. 1, pp. 20–31, 1995.
  • Z. S. Yu and D. G. Pu, “A new nonmonotone line search technique for unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 219, no. 1, pp. 134–144, 2008.
  • Z. S. Yu, C. X. He, and Y. Tian, “Global and local convergence of a nonmonotone trust region algorithm for equality constrained optimization,” Applied Mathematical Modelling, vol. 34, no. 5, pp. 1194–1202, 2010.
  • M. Ulbrich and S. Ulbrich, “Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function,” Mathematical Programming, vol. 95, no. 1, pp. 103–135, 2003.
  • 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.
  • N. Gould and P. L. Toint, “Global convergence of a non-monotone trust-region filter algorithm for nonlinear programming,” in Proceedings of Gainesville Conference on Multilevel Optimization, W. Hager, Ed., pp. 129–154, Kluwer, Dordrecht, The Netherlands, 2005.
  • K. Su and D. G. 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.
  • M. J. D. Powell, “Convergence properties of a class of minimization algorithm,” in Nonlinear Programming 2, O. Margasarian, R. Meyer, and S. Robinson, Eds., pp. 1–27, Academic Press, New York, NY, USA, 1975.
  • W. Hock and K. Schittkowski, Test Examples for Nonlinear Programming Codes, vol. 187 of Lecture Notes in Economics and Mathematical Systems, Springer, 1981.
  • K. Schittkowski, More Test Examples for Nonlinear Programming Codes, vol. 282 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1987.
  • K. Su and H. An, “Global convergence of a nonmonotone filter method for equality constrained optimization,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9396–9404, 2012.