Mathematical Programming

, Volume 119, Issue 1, pp 1–32 | Cite as

An affine-scaling interior-point CBB method for box-constrained optimization

  • William W. Hager
  • Bernard A. Mair
  • Hongchao Zhang
FULL LENGTH PAPER

Abstract

We develop an affine-scaling algorithm for box-constrained optimization which has the property that each iterate is a scaled cyclic Barzilai–Borwein (CBB) gradient iterate that lies in the interior of the feasible set. Global convergence is established for a nonmonotone line search, while there is local R-linear convergence at a nondegenerate local minimizer where the second-order sufficient optimality conditions are satisfied. Numerical experiments show that the convergence speed is insensitive to problem conditioning. The algorithm is particularly well suited for image restoration problems which arise in positron emission tomography where the cost function can be infinite on the boundary of the feasible set.

Keywords

Interior-point Affine-scaling Cyclic Barzilai–Borwein methods CBB PET Image reconstruction Global convergence Local convergence 

Mathematics Subject Classfication (2000)

90C06 90C26 65Y20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akaike H. (1959). On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Stat. Math. Tokyo 11: 1–17 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barzilai J. and Borwein J.M. (1988). Two point step size gradient methods. IMA J. Numer. Anal. 8: 141–148 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Birgin E.G., Biloti R., Tygel M. and Santos L.T. (1999). Restricted optimization: a clue to a fast and accurate implementation of the common reflection surface stack method. J. Appl. Geophys. 42: 143–155 CrossRefGoogle Scholar
  4. 4.
    Birgin E.G., Chambouleyron I. and Martínez J.M. (1999). Estimation of the optical constants and the thickness of thin films using unconstrained optimization. J. Comput. Phys. 151: 862–880 MATHCrossRefGoogle Scholar
  5. 5.
    Birgin E.G., Martínez J.M. and Raydan M. (2000). Nonmonotone spectral projected gradient methods for convex sets. SIAM J. Optim. 10: 1196–1211 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chang, J.-H., Anderson, J.M.M., Mair, B.A.: An accelerated penalized maximum likelihood algorithm for positron emission tomography. IEEE Trans. Nucl. Sci. 54(5): 1648–1659Google Scholar
  7. 7.
    Chang J.-H., Anderson J.M.M. and Votaw J.R. (2004). Regularized image reconstruction algorithms for positron emission tomography. IEEE Trans. Med. Imag. 23: 1165–1195 CrossRefGoogle Scholar
  8. 8.
    Coleman T.F. and Li Y. (1994). On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds. Math. Prog. 67: 189–224 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Coleman T.F. and Li Y. (1996). An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6: 418–445 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Coleman T.F. and Li Y. (2000). A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints. Math. Program. 88: 1–31 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dai Y.H. (2003). Alternate stepsize gradient method. Optimization 52: 395–415 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dai Y.H. and Fletcher R. (2005). Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100: 21–47 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dai Y.H., Hager W.W., Schittkowski K. and Zhang H. (2006). The cyclic Barzilai–Borwein method for unconstrained optimization. IMA J. Numer. Anal. 26: 604–627 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dai Y.H. and Zhang H. (2001). An adaptive two-point stepsize gradient algorithm. Numer. Algorithms 27: 377–385 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dikin I.I. (1967). Iterative solution of problems of linear and quadratic programming. Sov. Math. Dokl. 8: 674–675 MATHGoogle Scholar
  16. 16.
    Dostál Z. (1997). Box constrained quadratic programming with proportioning and projections. SIAM J. Optim. 7: 871–887 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fletcher, R.: On the Barzilai–Borwein method. Technical Report, Department of Mathematics, University of Dundee, Dundee (2001)Google Scholar
  18. 18.
    Friedlander A., Martínez J.M., Molina B. and Raydan M. (1999). Gradient method with retards and generalizations. SIAM J. Numer. Anal. 36: 275–289 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Glunt W., Hayden T.L. and Raydan M. (1993). Molecular conformations from distance matrices. J. Comput. Chem. 14: 114–120 CrossRefGoogle Scholar
  20. 20.
    Hager W.W. and Zhang H. (2006). A new active set algorithm for box constrained optimization. SIAM J. Optim. 17: 526–557 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Heinkenschloss M., Ulbrich M. and Ulbrich S. (1999). Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumption. Math. Program. 86: 615–635 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Johnson C.A., Seidel J. and Sofer A. (2000). Interior-point methodology for 3-D PET reconstruction. IEEE Trans. Med. Imag. 19: 271–285 CrossRefGoogle Scholar
  23. 23.
    Kanzow C. and Klug A. (2006). On affine-scaling interior-point Newton methods for nonlinear minimization with bound constraints. Comput. Optim. Appl. 35: 177–197 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kaufman L. (1987). Implementing and accelerating the EM algorithm for positron emission tomography. IEEE Trans. Med. Imag. 6: 37–51 CrossRefGoogle Scholar
  25. 25.
    Kaufman L. (1993). Maximum likelihood, least squares and penalized least squares for PET. IEEE Trans. Med. Imag. 12: 200–214 CrossRefGoogle Scholar
  26. 26.
    Raydan M. (1997). The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7: 26–33 MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Zhang H. and Hager W.W. (2004). A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14: 1043–1056 MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Zhang H. and Hager W.W.  (2005). PACBB: a projected adaptive cyclic Barzilai–Borwein method for box constrained optimization. In: Hager, W.W., Huang, S.-J., Pardalos, P.M. and Prokopyev, O.A. (eds) Multiscale Optimization Methods and Applications, pp 387–392. Springer, New York Google Scholar
  29. 29.
    Zhang, Y.: Interior-point gradient methods with diagonal-scalings for simple-bound constrained optimization. Technical Report, TR04-06, Department of Computational and Applied Mathematics. Rice University, Houston (2004)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • William W. Hager
    • 1
  • Bernard A. Mair
    • 1
  • Hongchao Zhang
    • 2
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Institute for Mathematics and its Applications (IMA)University of MinnesotaMinneapolisUSA

Personalised recommendations