Applied Mathematics and Optimization

, Volume 58, Issue 2, pp 275–290 | Cite as

Subspace Barzilai-Borwein Gradient Method for Large-Scale Bound Constrained Optimization

  • Yunhai XiaoEmail author
  • Qingjie Hu


An active set subspace Barzilai-Borwein gradient algorithm for large-scale bound constrained optimization is proposed. The active sets are estimated by an identification technique. The search direction consists of two parts: some of the components are simply defined; the other components are determined by the Barzilai-Borwein gradient method. In this work, a nonmonotone line search strategy that guarantees global convergence is used. Preliminary numerical results show that the proposed method is promising, and competitive with the well-known method SPG on a subset of bound constrained problems from CUTEr collection.


Bound constrained problem Barzilai-Borwein gradient method Nonmonotone line search Stationary point 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2007) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111, 5–32 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barzilai, J., Borwein, J.M.: Two point step size gradient method. IMA J. Numer. Anal. 8, 141–148 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertsekas, D.P.: Projected Newton methods for optimization problems with simple constrains. SIAM J. Control. Optim. 20, 221–246 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Birgin, E.G., Martínez, J.M.: Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput. Optim. Appl. 22, 101–125 (2002) CrossRefGoogle Scholar
  6. 6.
    Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1121 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Birgin, E.G., Martínez, J.M., Raydan, M.: Algorithm 813: SPG—software for convex-constrained optimization. ACM Trans. Math. Softw. 27, 340–349 (2001) zbMATHCrossRefGoogle Scholar
  8. 8.
    Birgin, E.G., Martínez, J.M., Raydan, M.: Inexact spectral projected gradient methods on convex sets. IMA J. Numer. Anal. 23, 539–559 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Burdakov, O.P., Martínez, J.M., Pilotta, E.A.: A limited memory multipoint secant method for bound constrained optimization. Annal. Oper. Res. 117, 51–70 (2002) zbMATHCrossRefGoogle Scholar
  10. 10.
    Byrd, R.H., Lu, P.H., Nocedal, J.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Stat. Comput. 16, 1190–1208 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Global convergence of a class of trust region algorithm for optimization with simple bounds. SIAM J. Numer. Anal. 25, 433–460 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Conn, A.R., Gould, N.I.M., Toint, Ph.L.: A globally convergent augmented Lagrangean algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28, 545–572 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Conn, A.R., Gould, N.I.M., Toint, Ph.L.: CUTE: constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21, 123–160 (1995) zbMATHCrossRefGoogle Scholar
  14. 14.
    Dai, Y.H., Fletcher, R.: Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100, 21–47 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dai, Y.H., Fletcher, R.: On the asymptotic behaviour of some new gradient methods. Math. Program. (Ser. A) 13, 541–559 (2005) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Dai, Y.H., Liao, L.Z.: R-linear convergence of the Barzilai and Borwein gradient method. IMA J. Numer. Anal. 26, 1–10 (2006) MathSciNetGoogle Scholar
  17. 17.
    Dai, Y.H., Yuan, J.Y., Yuan, Y.: Modified two-point stepsize gradient methods for unconstrained optimization. Comput. Optim. Appl. 22, 103–109 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dai, Y.H., Hager, W.W., Schittkowski, K., Zhang, H.: The cyclic Barzilai-Borwein method for unconstrained optimization. IMA J. Numer. Anal. 26, 604–627 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Diniz-Ehrhardt, M.A., Gomes-Ruggiero, M.A., Martínez, J.M., Santos, S.A.: Augmented Lagrangian algorithm based on the spectral projected gradient for solving nonlinear programming problems. J. Optim. Theory Appl. 123, 497–517 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Facchinei, F., Júdice, J., Soares, J.: An active set Newton’s algorithm for large-scale nonlinear programs with box constraints. SIAM J. Optim. 8, 158–186 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Facchinei, F., Lucidi, S., Palagi, L.: A truncated Newton algorithm for large scale box constrained optimization. SIAM J. Optim. 12, 1100–1125 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Grippo, L., Sciandrone, M.: Nonmonotone globalization techniques for the Barzilai-Borwein gradient method. Comput. Optim. Appl. 23, 143–169 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Goldfarb, D.: Extension of Davidon’s variable metric algorithm to maximization under linear inequality and constraints. SIAM J. Appl. Math. 17, 739–764 (1969) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Gould, N.I.M., Orban, D., L, Ph.: Toint, Numerical methods for large-scale nonlinear optimization. Acta Numer. 14, 299–361 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Hager, W.W., Zhang, H.: A new active set algorithm for box constrained optimization. SIAM J. Optim. 17, 526–557 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Krejić, N., Martínez, J.M., Mello, M.P., Pilotta, E.A.: Validation of an augmented Lagrangian algorithm with a Gauss-Newton Hessian approximation using a set of hard-spheres problems. Comput. Optim. Appl. 16, 247–263 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Luenberger, D.G.: Introduction to Linear and Nonlinear Programming, 2nd edn. Addison–Wesley, Reading (1986) Google Scholar
  30. 30.
    Martínez, J.M.: BOX-QUACAN and the implementation of Augmented Lagrangian algorithms for minimization with inequality constraints. Comput. Appl. Math. 19, 31–56 (2000) zbMATHMathSciNetGoogle Scholar
  31. 31.
    Ni, Q., Yuan, Y.: A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization. Math. Comput. 66, 1509–1520 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Raydan, M.: On the Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. IMA J. Numer. Anal. 13, 321–326 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Raydan, M.: The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7, 26–33 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14, 1043–1056 (2004) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of Applied Mathematics, School of Mathematics and Information ScienceHenan UniversityKaifengPeople’s Republic of China
  2. 2.Department of InformationHunan Business CollegeChangshaPeople’s Republic of China

Personalised recommendations