Abstract
A class of new affine-scaling interior-point Newton-type methods are considered for the solution of optimization problems with bound constraints. The methods are shown to be locally quadratically convergent under the strong second order sufficiency condition without assuming strict complementarity of the solution. The new methods differ from previous ones by Coleman and Li [Mathematical Programming, 67 (1994), pp. 189–224] and Heinkenschloss, Ulbrich, and Ulbrich [Mathematical Programming, 86 (1999), pp. 615–635] mainly in the choice of the scaling matrix. The scaling matrices used here have stronger smoothness properties and allow the application of standard results from non smooth analysis in order to obtain a relatively short and elegant local convergence result. An important tool for the definition of the new scaling matrices is the correct identification of the degenerate indices. Some illustrative numerical results with a comparison of the different scaling techniques are also included.
Similar content being viewed by others
References
S. Bellavia, M. Macconi, and B. Morini, “An affine scaling trust-region approach to bound-constrained nonlinear systems,” Applied Numerical Mathematics, vol. 44, pp. 257–280, 2003.
S. Bellavia, M. Macconi, and B. Morini, “STRSCNE: A scaled trust-region solver for constrained nonlinear systems,” Computational Optimization and Applications, vol. 28, pp. 31–50, 2004.
S. Bellavia and B. Morini, “An interior global method for nonlinear systems with simple bounds,” Optimization Methods and Software, to appear.
D.P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints,” SIAM Journal on Control and Optimization, vol. 20, pp. 221–246, 1982.
R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM Journal on Scientific and Statistical Computing, vol. 16, pp. 1190–1208, 1995.
F. H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York, 1983.
T. F. Coleman and Y. Li, “On the convergence of interior reflective Newton methods for nonlinear minimization subject to bounds,” Mathematical Programming, vol. 67, pp. 189–224, 1994.
T. F. Coleman and Y. Li, “An interior trust region approach for nonlinear minimization subject to bounds,” SIAM Journal on Optimization, vol. 6, pp. 418–445, 1996.
A.R. Conn, N.I.M. Gould, and Ph.L. Toint, “Global convergence of a class of trust region algorithms for optimization with simple bounds,” SIAM Journal on Numerical Analysis, vol. 25, pp. 433–460 1988. (Correction in: SIAM Journal on Numerical Analysis, vol. 26, pp. 764–767 1989).
A.R. Conn, N.I.M. Gould, and Ph.L. Toint, “Testing a class of methods for solving minimization problems with simple bounds on the variables,” Mathematics of Computation, vol. 50, pp. 399–430, 1988.
J.E. Dennis and L.N. Vicente, “Trust-region interior-point algorithms for minimization problems with simple bounds,” in H. Fischer, B. Riedmüller, and S. Schäffler (eds.), Applied Mathematics and Parallel Computing. Festschrift for Klaus Ritter. Physica, Heidelberg, 1996, pp. 97–107.
G. Di Pillo, “Exact penalty methods,” in: E. Spedicato (ed.), Algorithms for Continuous Optimization, The State of the Art. Kluwer, pp. 209–254, 1994.
G. Di Pillo and L. Grippo, “A continuously differentiable exact penalty function for nonlinear programming problems with inequality constraints,” SIAM Journal on Control and Optimization, vol. 23, pp. 72–84, 1985.
F. Facchinei, A. Fischer, and C. Kanzow, “On the accurate identification of active constraints,” SIAM Journal on Optimization, vol. 9, pp. 14–32, 1999.
F. Facchinei, J. Júdice, and J. Soares, “An active set Newton algorithm for large-scale nonlinear programs with box constraints,” SIAM Journal on Optimization, vol. 8, pp. 158–186, 1998.
F. Facchinei, S. Lucidi, and L. Palagi, “A truncated Newton algorithm for large scale box constrained optimization,” SIAM Journal on Optimization, vol. 12, pp. 1100–1125, 2002.
F. Facchinei and J.-S. Pang, “Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I,” Springer, New York-Berlin-Heidelberg, 2003.
F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume II. Springer, New York-Berlin-Heidelberg, 2003.
A. Fischer, “Solution of monotone complementarity problems with locally Lipschitzian functions,” Mathematical Programming, vol. 76, pp. 513–532, 1997.
A. Friedlander, J.M. Martínez, and S.A. Santos, “A new trust region algorithm for bound constrained minimization,” Applied Mathematics and Optimization, vol. 30, pp. 235–266, 1994.
M. Heinkenschloss, M. Ulbrich, and S. Ulbrich, “Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict comlpementarity assumption,” Mathematical Programming, vol. 86, pp. 615–635, 1999.
C. Kanzow, “Strictly feasible equation-based methods for mixed complementarity problems,” Numerische Mathematik, vol. 89, pp. 135–160, 2001.
M. Lescrenier, “Convergence of trust region algorithms for optimization with bounds when strict complementarity does not hold,” SIAM Journal on Numerical Analysis, vol. 28, pp. 476–495, 1991.
R.M. Lewis and V. Torczon, “Pattern search algorithms for bound constrained minimization,” SIAM Journal on Optimization, vol. 9, pp. 1082–1099, 1999.
C.-J. Lin and J.J. Moré, “Newton’s method for large bound-constrained optimization problems,” SIAM Journal on Optimization, vol. 9, pp. 1100–1127, 1999.
L. Qi, “Convergence analysis of some algorithms for solving nonsmooth equations,” Mathematics of Operations Research, vol. 18, pp. 227–244, 1993.
L. Qi and J. Sun, “A nonsmooth version of Newton’s method,” Mathematical Programming, vol. 58, pp. 353–367, 1993.
S.M. Robinson, “Strongly regular generalized equations,” Mathematics of Operations Research, vol. 5, pp. 43–62, 1980.
M. Ulbrich, S. Ulbrich, and M. Heinkenschloss, “Global convergence of affine-scaling interior-point Newton methods for infinite-dimensional nonlinear problems with pointwise bounds,” SIAM Journal on Control and Optimization, vol. 37, pp. 731–764, 1999.
M. Ulbrich and S. Ulbrich, “Superlinear convergence of affine-scaling interior-point Newton methods for infinite-dimensional nonlinear problems with pointwise bounds,” SIAM Journal on Control and Optimization, vol. 38, pp. 1938–1984, 2000.
C. Zhu, R. H. Byrd, and J. Nocedal, “Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization,” ACM Transactions on Mathematical Software, vol. 23, pp. 550–560, 1997.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kanzow, C., Klug, A. On Affine-Scaling Interior-Point Newton Methods for Nonlinear Minimization with Bound Constraints. Comput Optim Applic 35, 177–197 (2006). https://doi.org/10.1007/s10589-006-6514-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-006-6514-5