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Year 2023, Volume: 6 Issue: 3, 137 - 146, 30.09.2023
https://doi.org/10.33401/fujma.1261409

Abstract

References

  • [1] Z. H. Huang, Y. Zhang, W. Wu, A smoothing-type algorithm for solving system of inequalities, J. Comput. Appl. Math., 220 (2008), 355–363.
  • [2] Y. Liuyang, C. Fei, W. Zhongping, L. Weigang, W. Wenbo, A nonmonotone smoothing Newton method for system of nonlinear inequalities based on a new smoothing function, Comput Appl. Math., 134 (2019), 38–91.
  • [3] C. Wu, J. Wang, J. H. Alcantara and J. S. Chen, Smoothing strategy along with conjugate gradient algorithm for signal seconstruction, J. Sci. Comput. 87 (2021), 21.
  • [4] Y. Xiao, Q. Wang, Q. Hu, Non-smooth equations based method for `1-norm problems with applications to compressed sensing, Nonlinear Anal. Theory Methods Appl., 74(11) (2011), 3570–3577.
  • [5] A. S. Halilu, A. Majumder, M. Y. Waziri, K. Ahmed, A.M Awwal, Motion control of the two joint planar robotic manipulators through accelerated Dai–Liao method for solving system of nonlinear equations, Eng. Comput., 39(5) (2022), 1802–1840.
  • [6] D. Malyshev, L. Rybak, G. Carbone, T. Semenenko, A. Nozdracheva, Optimal design of a parallel manipulator for aliquoting of biomaterials considering workspace and singularity zones, Appl. Sci., 12(4) (2022), 2070.
  • [7] Y. Zhang, Z.-H. Huang, A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities, J. Comput. Appl. Math., 233 (2010) 2312–2321.
  • [8] J. S. Chen, C. H. Ko, Y. D. Liu, S. P. Wang, New smoothing functions for solving a system of equalities and inequalities, Pacific J. Optim, 12(1) (2016), 185-206.
  • [9] G. Yuan, T. Li, W. Hu, A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems, Appl. Numer. Math., 147 (2020), 129–141.
  • [10] R. Li, M. Cao, G. Zhou, A new adaptive accelerated Levenberg–Marquardt method for solving nonlinear equations and its applications in supply chain problems, Symmetry, 15(3) (2023), 588.
  • [11] D.Q. Mayne, E. Polak, A. J. Heunis, Solving nonlinear inequalities in a finite number of iterations, J. Optim. Theory Appl., 33(2) (1981), 207–221.
  • [12] H. Che, M. Li, A smoothing and regularization Broyden-like method for nonlinear inequalities, J. Appl. Math. Comput., 41 (2013) 209–227.
  • [13] A. B. Abubakar, P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numer. Algorithms, 81 (2019), 197-210.
  • [14] M. Y. Waziri, K. Ahmed, K. Two descent Dai-Yuan conjugate gradient methods for systems of monotone nonlinear equations, J. Sci. Comput., 90 (2022), 1-53.
  • [15] Z. Saeidian, M. R. Peyghami, M. Habibi, S. Ghasemi, A new trust-region method for solving systems of equalities and inequalities, Comp. Appl. Math. 36 (2017), 769 70.
  • [16] X. Fan, Q. Yan, Solving system of inequalities via a smoothing homotopy method, Numer Algorithms 82 (2019), 719–728.
  • [17] L. Qi, D. Sun, Smoothing functions and smoothing Newton method for complementarity and variational inequality problems, J. Optim. Theory Appl. 113 (2002), 121–147.
  • [18] S. Huang, Z. Wan, X. H. Chen, A new nonmonotone line search technique for unconstrained optimization, Numer. Algorithms 68 (4) (2015), 671–689.
  • [19] H. C. Zhang, W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim., 14(4) (2004), 1043–756.
  • [20] X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program. Ser. (B), 134 (2012), 71–99.
  • [21] D. Bertsekas, Nondifferentiable optimization via approximation, Math Programmming Study, 3 (1975) 1–25.
  • [22] I. Zang, A smooting out technique for min-max optimization, Math. Programm., 19 (1980), 61–77.
  • [23] C. Chen, O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problem, Comput. Optim. Appl., 5 (1996) 97-138.
  • [24] A. E. Xavier, Optimal covering of plane domains by circles via hyperbolic smoothing, J. Global Optim., 31 (2005), 493–504.
  • [25] A. M. Bagirov, A. Al Nuamiat, N. Sultanova, Hyperbolic smoothing functions for nonsmooth minimization, Optimization, 62 (6) (2013), 759-782.
  • [26] A. Sahiner, G. Kapusuz, N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Control Optim., 6 (2) (2016) 161–173.
  • [27] N. Yilmaz, A. Sahiner, On a new smoothing technique for non-smooth, non-convex optimization, Numer. Algebra Control Optim., 10 (2020), 317–330.
  • [28] C. Grossmann, Smoothing techniques for exact penalty function methods, Contemporary Mathematics: A Panaroma of Mathematics Pure and Applied, American Mathematical Society, Rhode Island, 658 (2016) 249–265.
  • [29] L. Qi, P. Tseng, On almost smooth functions and piecewise smooth functions, Nonlinear Anal. 67 (2007) 773–794.
  • [30] N. Yilmaz, A. Sahiner, New smoothing approximations to piecewise smooth functions and applications, Numer. Func. Anal. Optim., 40 (2019), 513–534.

A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities

Year 2023, Volume: 6 Issue: 3, 137 - 146, 30.09.2023
https://doi.org/10.33401/fujma.1261409

Abstract

In this study, the system of nonlinear inequalities (SNI) problem is investigated. First, a SNI is reformulated as a system of nonsmooth and nonlinear equations (SNNE). Second, a new smoothing technique for the "$\max$" function is proposed and the smoothing technique is employed for each element of the SNNE. Then, a new smoothing algorithm is developed in order to solve SNNE by combining the smoothing technique with the iterative method. The new algorithm is applied to some numerical examples to show the efficiency of our algorithm.

References

  • [1] Z. H. Huang, Y. Zhang, W. Wu, A smoothing-type algorithm for solving system of inequalities, J. Comput. Appl. Math., 220 (2008), 355–363.
  • [2] Y. Liuyang, C. Fei, W. Zhongping, L. Weigang, W. Wenbo, A nonmonotone smoothing Newton method for system of nonlinear inequalities based on a new smoothing function, Comput Appl. Math., 134 (2019), 38–91.
  • [3] C. Wu, J. Wang, J. H. Alcantara and J. S. Chen, Smoothing strategy along with conjugate gradient algorithm for signal seconstruction, J. Sci. Comput. 87 (2021), 21.
  • [4] Y. Xiao, Q. Wang, Q. Hu, Non-smooth equations based method for `1-norm problems with applications to compressed sensing, Nonlinear Anal. Theory Methods Appl., 74(11) (2011), 3570–3577.
  • [5] A. S. Halilu, A. Majumder, M. Y. Waziri, K. Ahmed, A.M Awwal, Motion control of the two joint planar robotic manipulators through accelerated Dai–Liao method for solving system of nonlinear equations, Eng. Comput., 39(5) (2022), 1802–1840.
  • [6] D. Malyshev, L. Rybak, G. Carbone, T. Semenenko, A. Nozdracheva, Optimal design of a parallel manipulator for aliquoting of biomaterials considering workspace and singularity zones, Appl. Sci., 12(4) (2022), 2070.
  • [7] Y. Zhang, Z.-H. Huang, A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities, J. Comput. Appl. Math., 233 (2010) 2312–2321.
  • [8] J. S. Chen, C. H. Ko, Y. D. Liu, S. P. Wang, New smoothing functions for solving a system of equalities and inequalities, Pacific J. Optim, 12(1) (2016), 185-206.
  • [9] G. Yuan, T. Li, W. Hu, A conjugate gradient algorithm for large-scale nonlinear equations and image restoration problems, Appl. Numer. Math., 147 (2020), 129–141.
  • [10] R. Li, M. Cao, G. Zhou, A new adaptive accelerated Levenberg–Marquardt method for solving nonlinear equations and its applications in supply chain problems, Symmetry, 15(3) (2023), 588.
  • [11] D.Q. Mayne, E. Polak, A. J. Heunis, Solving nonlinear inequalities in a finite number of iterations, J. Optim. Theory Appl., 33(2) (1981), 207–221.
  • [12] H. Che, M. Li, A smoothing and regularization Broyden-like method for nonlinear inequalities, J. Appl. Math. Comput., 41 (2013) 209–227.
  • [13] A. B. Abubakar, P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numer. Algorithms, 81 (2019), 197-210.
  • [14] M. Y. Waziri, K. Ahmed, K. Two descent Dai-Yuan conjugate gradient methods for systems of monotone nonlinear equations, J. Sci. Comput., 90 (2022), 1-53.
  • [15] Z. Saeidian, M. R. Peyghami, M. Habibi, S. Ghasemi, A new trust-region method for solving systems of equalities and inequalities, Comp. Appl. Math. 36 (2017), 769 70.
  • [16] X. Fan, Q. Yan, Solving system of inequalities via a smoothing homotopy method, Numer Algorithms 82 (2019), 719–728.
  • [17] L. Qi, D. Sun, Smoothing functions and smoothing Newton method for complementarity and variational inequality problems, J. Optim. Theory Appl. 113 (2002), 121–147.
  • [18] S. Huang, Z. Wan, X. H. Chen, A new nonmonotone line search technique for unconstrained optimization, Numer. Algorithms 68 (4) (2015), 671–689.
  • [19] H. C. Zhang, W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim., 14(4) (2004), 1043–756.
  • [20] X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Program. Ser. (B), 134 (2012), 71–99.
  • [21] D. Bertsekas, Nondifferentiable optimization via approximation, Math Programmming Study, 3 (1975) 1–25.
  • [22] I. Zang, A smooting out technique for min-max optimization, Math. Programm., 19 (1980), 61–77.
  • [23] C. Chen, O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problem, Comput. Optim. Appl., 5 (1996) 97-138.
  • [24] A. E. Xavier, Optimal covering of plane domains by circles via hyperbolic smoothing, J. Global Optim., 31 (2005), 493–504.
  • [25] A. M. Bagirov, A. Al Nuamiat, N. Sultanova, Hyperbolic smoothing functions for nonsmooth minimization, Optimization, 62 (6) (2013), 759-782.
  • [26] A. Sahiner, G. Kapusuz, N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Control Optim., 6 (2) (2016) 161–173.
  • [27] N. Yilmaz, A. Sahiner, On a new smoothing technique for non-smooth, non-convex optimization, Numer. Algebra Control Optim., 10 (2020), 317–330.
  • [28] C. Grossmann, Smoothing techniques for exact penalty function methods, Contemporary Mathematics: A Panaroma of Mathematics Pure and Applied, American Mathematical Society, Rhode Island, 658 (2016) 249–265.
  • [29] L. Qi, P. Tseng, On almost smooth functions and piecewise smooth functions, Nonlinear Anal. 67 (2007) 773–794.
  • [30] N. Yilmaz, A. Sahiner, New smoothing approximations to piecewise smooth functions and applications, Numer. Func. Anal. Optim., 40 (2019), 513–534.
There are 30 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Nurullah Yılmaz 0000-0001-6429-7518

Ayşegül Kayacan 0000-0002-6285-5771

Publication Date September 30, 2023
Submission Date March 9, 2023
Acceptance Date June 19, 2023
Published in Issue Year 2023 Volume: 6 Issue: 3

Cite

APA Yılmaz, N., & Kayacan, A. (2023). A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities. Fundamental Journal of Mathematics and Applications, 6(3), 137-146. https://doi.org/10.33401/fujma.1261409
AMA Yılmaz N, Kayacan A. A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities. FUJMA. September 2023;6(3):137-146. doi:10.33401/fujma.1261409
Chicago Yılmaz, Nurullah, and Ayşegül Kayacan. “A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities”. Fundamental Journal of Mathematics and Applications 6, no. 3 (September 2023): 137-46. https://doi.org/10.33401/fujma.1261409.
EndNote Yılmaz N, Kayacan A (September 1, 2023) A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities. Fundamental Journal of Mathematics and Applications 6 3 137–146.
IEEE N. Yılmaz and A. Kayacan, “A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities”, FUJMA, vol. 6, no. 3, pp. 137–146, 2023, doi: 10.33401/fujma.1261409.
ISNAD Yılmaz, Nurullah - Kayacan, Ayşegül. “A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities”. Fundamental Journal of Mathematics and Applications 6/3 (September 2023), 137-146. https://doi.org/10.33401/fujma.1261409.
JAMA Yılmaz N, Kayacan A. A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities. FUJMA. 2023;6:137–146.
MLA Yılmaz, Nurullah and Ayşegül Kayacan. “A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities”. Fundamental Journal of Mathematics and Applications, vol. 6, no. 3, 2023, pp. 137-46, doi:10.33401/fujma.1261409.
Vancouver Yılmaz N, Kayacan A. A New Smoothing Algorithm to Solve a System of Nonlinear Inequalities. FUJMA. 2023;6(3):137-46.

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