A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems

Min Xi; Ailing Xiao

doi:doi:10.11648/j.ajam.20180602.17

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A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems

Min Xi, Ailing Xiao

Published in American Journal of Applied Mathematics (Volume 6, Issue 2)

Received: 25 June 2018 Published: 26 June 2018

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Abstract

A class of nonsmooth composite minimization problems are considered in this paper. In practice, many problems in computational science, engineering and other enormous areas fall into this class of problems. Two obvious applications of this class of problems are least square problems with nonsmooth data and the problem of solving a system of nonsmooth equations. Because of its practical importance, this class of problem has received wide attention from the mathematical optimization community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for this class of problems and also provided the convergence analysis to support their algorithm. Motivated partly by their work, in this paper a nonmonotone trust region algorithm for this class of nonsmooth composite minimization problems is presented. Different from most existing monotone line search and trust region methods, this method combines the nonmonotone technique to improve the efficiency of the trust region method. After a brief introduction of the class of problems in the first section, some fundamental concepts and properties which will be used in this paper are presented. Then, the new nonmonotone trust region algorithm for the class of problem is described followed by the global convergence analysis of the new algorithm. A simple application of this algorithm is discussed in the last part of this paper.

Published in	American Journal of Applied Mathematics (Volume 6, Issue 2)
DOI	10.11648/j.ajam.20180602.17
Page(s)	71-77
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Nonmonotone, Trust Region Method, Composite, Nonsmooth Optimization

References

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APA Style

Min Xi, Ailing Xiao. (2018). A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems. American Journal of Applied Mathematics, 6(2), 71-77. https://doi.org/10.11648/j.ajam.20180602.17

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ACS Style

Min Xi; Ailing Xiao. A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems. Am. J. Appl. Math. 2018, 6(2), 71-77. doi: 10.11648/j.ajam.20180602.17

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AMA Style

Min Xi, Ailing Xiao. A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems. Am J Appl Math. 2018;6(2):71-77. doi: 10.11648/j.ajam.20180602.17

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@article{10.11648/j.ajam.20180602.17,
  author = {Min Xi and Ailing Xiao},
  title = {A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {2},
  pages = {71-77},
  doi = {10.11648/j.ajam.20180602.17},
  url = {https://doi.org/10.11648/j.ajam.20180602.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180602.17},
  abstract = {A class of nonsmooth composite minimization problems are considered in this paper. In practice, many problems in computational science, engineering and other enormous areas fall into this class of problems. Two obvious applications of this class of problems are least square problems with nonsmooth data and the problem of solving a system of nonsmooth equations. Because of its practical importance, this class of problem has received wide attention from the mathematical optimization community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for this class of problems and also provided the convergence analysis to support their algorithm. Motivated partly by their work, in this paper a nonmonotone trust region algorithm for this class of nonsmooth composite minimization problems is presented. Different from most existing monotone line search and trust region methods, this method combines the nonmonotone technique to improve the efficiency of the trust region method. After a brief introduction of the class of problems in the first section, some fundamental concepts and properties which will be used in this paper are presented. Then, the new nonmonotone trust region algorithm for the class of problem is described followed by the global convergence analysis of the new algorithm. A simple application of this algorithm is discussed in the last part of this paper.},
 year = {2018}
}

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TY  - JOUR
T1  - A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems
AU  - Min Xi
AU  - Ailing Xiao
Y1  - 2018/06/26
PY  - 2018
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DO  - 10.11648/j.ajam.20180602.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
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EP  - 77
PB  - Science Publishing Group
SN  - 2330-006X
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AB  - A class of nonsmooth composite minimization problems are considered in this paper. In practice, many problems in computational science, engineering and other enormous areas fall into this class of problems. Two obvious applications of this class of problems are least square problems with nonsmooth data and the problem of solving a system of nonsmooth equations. Because of its practical importance, this class of problem has received wide attention from the mathematical optimization community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for this class of problems and also provided the convergence analysis to support their algorithm. Motivated partly by their work, in this paper a nonmonotone trust region algorithm for this class of nonsmooth composite minimization problems is presented. Different from most existing monotone line search and trust region methods, this method combines the nonmonotone technique to improve the efficiency of the trust region method. After a brief introduction of the class of problems in the first section, some fundamental concepts and properties which will be used in this paper are presented. Then, the new nonmonotone trust region algorithm for the class of problem is described followed by the global convergence analysis of the new algorithm. A simple application of this algorithm is discussed in the last part of this paper.
VL  - 6
IS  - 2
ER  -

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Author Information

Min Xi

School of Finance, Guangdong University of Foreign Studies, Guangzhou, China
Ailing Xiao

School of Finance, Guangdong University of Foreign Studies, Guangzhou, China

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Min Xi, Ailing Xiao. (2018). A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems. American Journal of Applied Mathematics, 6(2), 71-77. https://doi.org/10.11648/j.ajam.20180602.17

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ACS Style

Min Xi; Ailing Xiao. A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems. Am. J. Appl. Math. 2018, 6(2), 71-77. doi: 10.11648/j.ajam.20180602.17

Copy | Download

AMA Style

Min Xi, Ailing Xiao. A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems. Am J Appl Math. 2018;6(2):71-77. doi: 10.11648/j.ajam.20180602.17

Copy | Download

@article{10.11648/j.ajam.20180602.17,
  author = {Min Xi and Ailing Xiao},
  title = {A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems},
  journal = {American Journal of Applied Mathematics},
  volume = {6},
  number = {2},
  pages = {71-77},
  doi = {10.11648/j.ajam.20180602.17},
  url = {https://doi.org/10.11648/j.ajam.20180602.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180602.17},
  abstract = {A class of nonsmooth composite minimization problems are considered in this paper. In practice, many problems in computational science, engineering and other enormous areas fall into this class of problems. Two obvious applications of this class of problems are least square problems with nonsmooth data and the problem of solving a system of nonsmooth equations. Because of its practical importance, this class of problem has received wide attention from the mathematical optimization community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for this class of problems and also provided the convergence analysis to support their algorithm. Motivated partly by their work, in this paper a nonmonotone trust region algorithm for this class of nonsmooth composite minimization problems is presented. Different from most existing monotone line search and trust region methods, this method combines the nonmonotone technique to improve the efficiency of the trust region method. After a brief introduction of the class of problems in the first section, some fundamental concepts and properties which will be used in this paper are presented. Then, the new nonmonotone trust region algorithm for the class of problem is described followed by the global convergence analysis of the new algorithm. A simple application of this algorithm is discussed in the last part of this paper.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - A Nonmonotone Trust Region Method for Nonsmooth Composite Programming Problems
AU  - Min Xi
AU  - Ailing Xiao
Y1  - 2018/06/26
PY  - 2018
N1  - https://doi.org/10.11648/j.ajam.20180602.17
DO  - 10.11648/j.ajam.20180602.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 71
EP  - 77
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20180602.17
AB  - A class of nonsmooth composite minimization problems are considered in this paper. In practice, many problems in computational science, engineering and other enormous areas fall into this class of problems. Two obvious applications of this class of problems are least square problems with nonsmooth data and the problem of solving a system of nonsmooth equations. Because of its practical importance, this class of problem has received wide attention from the mathematical optimization community. In particular, Sampaio, Yuan and Sun has proposed a trust region method for this class of problems and also provided the convergence analysis to support their algorithm. Motivated partly by their work, in this paper a nonmonotone trust region algorithm for this class of nonsmooth composite minimization problems is presented. Different from most existing monotone line search and trust region methods, this method combines the nonmonotone technique to improve the efficiency of the trust region method. After a brief introduction of the class of problems in the first section, some fundamental concepts and properties which will be used in this paper are presented. Then, the new nonmonotone trust region algorithm for the class of problem is described followed by the global convergence analysis of the new algorithm. A simple application of this algorithm is discussed in the last part of this paper.
VL  - 6
IS  - 2
ER  -

Copy | Download