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Journal of Optimization Theory and Applications

, Volume 168, Issue 1, pp 129–152 | Cite as

A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations

  • Gonglin YuanEmail author
  • Zehong Meng
  • Yong Li
Article

Abstract

It is well known that nonlinear conjugate gradient methods are very effective for large-scale smooth optimization problems. However, their efficiency has not been widely investigated for large-scale nonsmooth problems, which are often found in practice. This paper proposes a modified Hestenes–Stiefel conjugate gradient algorithm for nonsmooth convex optimization problems. The search direction of the proposed method not only possesses the sufficient descent property but also belongs to a trust region. Under suitable conditions, the global convergence of the presented algorithm is established. The numerical results show that this method can successfully be used to solve large-scale nonsmooth problems with convex and nonconvex properties (with a maximum dimension of 60,000). Furthermore, we study the modified Hestenes–Stiefel method as a solution method for large-scale nonlinear equations and establish its global convergence. Finally, the numerical results for nonlinear equations are verified, with a maximum dimension of 100,000.

Keywords

Nonsmooth Nonlinear equations Conjugate gradient Large scale Global convergence 

Mathematics Subject Classification

65K05 90C26 

Notes

Acknowledgments

The authors thank the referees and the editor for their valuable comments which greatly improve our paper. The authors would like to thank Postdoctoral Researcher Yajun Xiao of the University of Technology, Sydney, for his assistance in editing this manuscript. This work was supported by the QFRC visitor funds of the University of Technology, Sydney; the study abroad funds for Guangxi talents in China; the Program for Excellent Talents in Guangxi Higher Education Institutions (Grant No. 201261); the Guangxi NSF (Grant No. 2012GXNSFAA053002); and the China NSF (Grant Nos. 11261006 and 11161003).

References

  1. 1.
    Kärkkäinen, T., Majava, K., Mäkelä, M.M.: Comparison of formulations and solution methods for image restoration problems. Inverse Probl. 17, 1977–1995 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Li, J., Li, X., Yang, B., Sun, X.: Segmentation-based image copy-move forgery detection scheme. IEEE Trans. Inf. Forensics Secur. 10, 507–518 (2015)CrossRefGoogle Scholar
  3. 3.
    Zhang, H., Wu, Q., Nguyen, T., Sun, X.: Synthetic aperture radar image segmentation by modified student’s t-mixture model. IEEE Trans. Geosci. Remote Sens. 52, 4391–4403 (2014)CrossRefGoogle Scholar
  4. 4.
    Mäkelä, M.M., Neittaanmäki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific Publishing Co., Singapore (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20, 33–61 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Birge, J.R., Qi, L., Wei, Z.: A general approach to convergence properties of some methods for nonsmooth convex optimization. Appl. Math. Optim. 38, 141–158 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Rockafellar, R.T.: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Birge, J.R., Qi, L., Wei, Z.: Convergence analysis of some methods for minimizing a nonsmooth convex function. J. Optim. Theory Appl. 97, 357–383 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bonnans, J.F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A.: A family of variable metric proximal methods. Math. Program. 68, 15–47 (1995)zbMATHGoogle Scholar
  10. 10.
    Wei, Z., Qi, L.: Convergence analysis of a proximal Newton method. Numer. Funct. Anal. Optim. 17, 463–472 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wei, Z., Qi, L., Birge, J.R.: A new methods for nonsmooth convex optimization. J. Inequal. Appl. 2, 157–179 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yuan, G., Wei, Z.: The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems. Math. Model. Anal. 17, 203–216 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sagara, N., Fukushima, M.: A trust region method for nonsmooth convex optimization. J. Ind. Manag. Optim. 1, 171–180 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yuan, G., Wei, Z., Wang, Z.: Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minimization. Comput. Optim. Appl. 54, 45–64 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lemaréchal, C.: Extensions diverses des méthodes de gradient et applications. Thèse d’Etat, Paris (1980)Google Scholar
  16. 16.
    Wolfe, P.: A method of conjugate subgradients for minimizing nondifferentiable convex functions. Math. Program. Stud. 3, 145–173 (1975)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable optimization. Math. Program. 46, 105–122 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2, 121–152 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, vol. 1133. Springer, Berlin (1985)Google Scholar
  20. 20.
    Kiwiel, K.C.: Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities. Math. Program. 69, 89–109 (1995)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Schramm, H.: Eine kombination yon bundle-und trust-region-verfahren zur Lösung nichtdifferenzierbare optimierungsprobleme. Bayreuther Mathematische Schriften, Heft 30. Universitat Bayreuth, Germany (1989)Google Scholar
  22. 22.
    Haarala, M., Miettinen, K., Mäkelä, M.M.: New limited memory bundle method for large-scale nonsmooth optimization. Optim. Methods Softw. 19, 673–692 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hestenes, M.R., Stiefel, E.: Method of conjugate gradient for solving linear equations. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fletcher, R., Reeves, C.: Function minimization by conjugate gradients. Comput. J. 7, 149–154 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dai, Y., Yuan, Y.: A nonlinear conjugate gradient with a strong global convergence properties. SIAM J. Optim. 10, 177–182 (2000)CrossRefGoogle Scholar
  26. 26.
    Fletcher, R.: Practical Method of Optimization, Vol I: Unconstrained Optimization, 2nd edn. Wiley, New York (1997)Google Scholar
  27. 27.
    Polak, E., Ribière, G.: Note sur la convergence de directions conjugees. Rev. Fr. Inform. Rech. Opér. 3, 35–43 (1969)zbMATHGoogle Scholar
  28. 28.
    Gilbert, J.C., Nocedal, J.: Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim. 2, 21–42 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hu, Y.F., Storey, C.: Global convergence result for conjugate method. J. Optim. Theory Appl. 71, 399–405 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wei, Z., Li, G., Qi, L.: Global convergence of the Polak–Ribière–Polyak conjugate gradient methods with inexact line search for nonconvex unconstrained optimization problems. Math. Comput. 77, 2173–2193 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ahmed, T., Storey, D.: Efficient hybrid conjugate gradient techniques. J. Optim. Theory Appl. 64, 379–394 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Al-Baali, A.: Descent property and global convergence of the Flecher–Reeves method with inexact line search. IMA J. Numer. Anal. 5, 121–124 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wei, Z., Yao, S., Liu, L.: The convergence properties of some new conjugate gradient methods. Appl. Math. Comput. 183, 1341–1350 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yuan, G.: Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems. Optim. Lett. 3, 11–21 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yuan, G., Lu, X.: A modified PRP conjugate gradient method. Ann. Oper. Res. 166, 73–90 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Yuan, G., Lu, X., Wei, Z.: A conjugate gradient method with descent direction for unconstrained optimization. J. Comput. Appl. Math. 233, 519–530 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zhang, L., Zhou, W., Li, D.: A descent modified Polak–Ribière–Polyak conjugate method and its global convergence. IMA J. Numer. Anal. 26, 629–649 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Buhmiler, S., Krejić, N., Lužanin, Z.: Practical quasi-Newton algorithms for singular nonlinear systems. Numer. Algorithms 55, 481–502 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Solodov, M.V., Svaiter, B.F.: A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 355–369. Kluwer Academic Publishers, Dordrecht (1998)CrossRefGoogle Scholar
  41. 41.
    Toint, P.L.: Numerical solution of large sets of algebraic nonlinear equations. Math. Comput. 173, 175–189 (1986)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Yuan, G., Lu, X.: A new backtracking inexact BFGS method for symmetric nonlinear equations. Comput. Math. Appl. 55, 116–129 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Yuan, G., Yao, S.: A BFGS algorithm for solving symmetric nonlinear equations. Optimization 62, 82–95 (2013)MathSciNetCrossRefGoogle Scholar
  44. 44.
    La Cruz, W., Martínez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1429–1448 (2006)CrossRefzbMATHGoogle Scholar
  45. 45.
    La Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18, 583–599 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Tong, X., Qi, L.: On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solutions. J. Optim. Theory Appl. 123, 187–211 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Yuan, G., Lu, X., Wei, Z.: BFGS trust-region method for symmetric nonlinear equations. J. Comput. Appl. Math. 230, 44–58 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Yuan, G., Wei, Z., Lu, X.: A BFGS trust-region method for nonlinear equations. Computing 92, 317–333 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Zhang, J., Wang, Y.: A new trust region method for nonlinear equations. Math. Methods Oper. Res. 58, 283–298 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Grippo, L., Sciandrone, M.: Nonmonotone derivative-free methods for nonlinear equations. Comput. Optim. Appl. 37, 297–328 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Yuan, Y.: Subspace methods for large scale nonlinear equations and nonlinear least squares. Optim. Eng. 10, 207–218 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Fasano, G., Lampariello, F., Sciandrone, M.: A truncated nonmonotone Gauss–Newton method for large-scale nonlinear least-squares problems. Comput. Optim. Appl. 34, 343–358 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Li, D., Fukushima, M.: A global and superlinear convergent Gauss–Newton-based BFGS method for symmetric nonlinear equations. SIAM J. Numer. Anal. 37, 152–172 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Li, D., Qi, L., Zhou, S.: Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J. Numer. Anal. 40, 1763–1774 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Kanzow, C., Yamashita, N., Fukushima, M.: Levenberg–Marquardt methods for constrained nonlinear equations with strong local convergence properties. J. Comput. Appl. Math. 172, 375–397 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431–441 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Bouaricha, A., Schnabel, R.B.: Tensor methods for large sparse systems of nonlinear equations. Math. Program. 82, 377–400 (1998)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Cheng, W.: A PRP type method for systems of monotone equations. Math. Comput. Model. 50, 15–20 (2009)CrossRefzbMATHGoogle Scholar
  59. 59.
    Yu, G., Guan, L., Chen, W.: Spectral conjugate gradient methods with sufficient descent property for large-scale unconstraned optimization. Optim. Methods Softw. 23, 275–293 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Yuan, G., Wei, Z., Lu, S.: Limited memory BFGS method with backtracking for symmetric nonlinear equations. Math. Comput. Model. 54, 367–377 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Wang, C.W., Wang, Y.J., Xu, C.L.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Methods Oper. Res. 66, 33–46 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Yu, Z.S., Lian, J., Sun, J., Xiao, Y.H., Liu, L., Li, Z.H.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59, 2416–2423 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Li, D., Wang, L.: A modified Fletcher–Reeves-type derivative-free method for symmetric nonlinear equations. Numer. Algebra Control Optim. 1, 71–82 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Yuan, G., Wei, Z., Zhao, Q.: A modified Polak–Ribière–Polyak conjugate gradient algorithm for large-scale optimization problems. IIE Trans. 46, 397–413 (2014)CrossRefGoogle Scholar
  65. 65.
    Yuan, G., Zhang, M.: A modified Hestenes–Stiefel conjugate gradient algorithm for large-scale optimization. Numer. Funct. Anal. Optim. 34, 914–937 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Zhang, L., Zhou, W.J.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196, 478–484 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Andrei, N.: Another hybrid conjugate gradient algorithm for unconstrained optimization. Numer. Algorithms 47, 143–156 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Li, Q., Li, D.: A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J. Numer. Anal. 31, 1625–1635 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 69.
    Fukushima, M., Qi, L.: A global and superlinearly convergent algorithm for nonsmooth convex minimization. SIAM J. Optim. 6, 1106–1120 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Correa, R., Lemaréchal, C.: Convergence of some algorithms for convex minimization. Math. Program. 62, 261–273 (1993)CrossRefzbMATHGoogle Scholar
  72. 72.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II. Springer, Berlin (1993)zbMATHGoogle Scholar
  73. 73.
    Calamai, P.H., Moré, J.J.: Projected gradient methods for linear constrained problems. Math. Program. 39, 93–116 (1987)CrossRefzbMATHGoogle Scholar
  74. 74.
    Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–245 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  76. 76.
    Yuan, G., Wei, Z.: A modified PRP conjugate gradient algorithm with nonmonotone line search for nonsmooth convex optimization problems. J. Appl. Math. Comput. (2011, in press)Google Scholar
  77. 77.
    Yuan, G., Wei, Z., Li, G.: A modified Polak–Ribière–Polyak conjugate gradient algorithm with nonmonotone line search for nonsmooth convex minimization. J. Comput. Appl. Math. 255, 86–96 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  78. 78.
    Lukšan, L., Vlšek, J.: Test problems for nonsmooth unconstrained and linearly constrained optimization. Technical Report No. 798. Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague (2000)Google Scholar
  79. 79.
    Lukšan, L., Vlšek, J.: A bundle-Newton method for nonsmooth unconstrained minimization. Math. Program. 83, 373–391 (1998)zbMATHGoogle Scholar
  80. 80.
    Polak, E.: The conjugate gradient method in extreme problems. Comput. Math. Math. Phys. 9, 94–112 (1969)CrossRefGoogle Scholar
  81. 81.
    Fukushima, M.: A descent algorithm for nonsmooth convex optimization. Math. Program. 30, 163–175 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  82. 82.
    Karmitsa, N., Bagirov, A., Mäkelä, M.M.: Comparing different nonsmooth minimization methods and software. Optim. Methods Softw. 27, 131–153 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Kappel, F., Kuntsevich, A.: An implementation of Shor’s r-algorithm. Comput. Optim. Appl. 15, 193–205 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Kuntsevich, A., Kappel, F.: SolvOpt-the Solver for Local Nonlinear Optimization Problems. Karl-Franzens University of Graz, Graz (1997)Google Scholar
  85. 85.
    Shor, N.Z.: Minimization Methods for Non-Differentiable Functions. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  86. 86.
    Mäkelä, M.M.: Multiobjective proximal bundle method for nonconvex nonsmooth optimization: Fortran subroutine MPBNGC 2.0. Reports of the Department of Mathematical Information Technology, Series B, Scientific Computing, No. B 13/2003, University of Jyväkylä, Jyväkylä (2003)Google Scholar
  87. 87.
    Haarala, M., Miettinen, K., Mäkelä, M.M.: Globally convergent limited memory bundle method for large-scale nonsmooth optimization. Math. Program. 109, 181–205 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    Bagirov, A.M., Karasozen, B., Sezer, M.: Discrete gradient method: a derivative free method for nonsmooth optimization. J. Optim. Theory Appl. 137, 317–334 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  89. 89.
    Bagirov, A.M., Ganjehlou, A.N.: A quasisecant method for minimizing nonsmooth functions. Optim. Methods Softw. 25, 3–18 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  91. 91.
    Moré, J., Garbow, B., Hillström, K.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981)CrossRefzbMATHGoogle Scholar
  92. 92.
    Solodov, M.V., Svaiter, B.F.: A hybrid projection-proximal point algorithm. J. Convex Anal. 6, 59–70 (1999)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.The Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications, College of Mathematics and Information ScienceGuangxi UniversityNanningChina
  2. 2.School of Mathematics and StatisticsZhejiang University of Finance and EconomicsHangzhouChina
  3. 3.Department of MathematicsBaise UniversityBaiseChina

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