Abstract.
The classical trust-region method for unconstrained minimization can be augmented with a line search that finds a point that satisfies the Wolfe conditions. One can use this new method to define an algorithm that simultaneously satisfies the quasi-Newton condition at each iteration and maintains a positive-definite approximation to the Hessian of the objective function. This new algorithm has strong global convergence properties and is robust and efficient in practice.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bongartz, I., Conn, A.R., Gould, N.I.M., Toint, Ph.L., CUTE: Constrained and unconstrained testing environment. Département de Mathématique Facultés Universitaires de Namur, 1993
Conn, A.R., Nicholas, I.M. Gould, Toint, P.L.: Trust-Region Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia PA, 2000
Fletcher, R.: An Algorithm for Solving Linearly Constrained Optimization Problems. Math. Program. 2, 133–165 (1972)
Fletcher, R.: Practical Methods of Optimization. John Wiley and Sons 1987
Friedlander, A., Martínez, J. M., Santos, S. A.: A new trust region algorithm for bound constrained minimization. Appl. Math. Optim. 30, 235–266 (1994)
Gill, P.E., Goluband, G.H., Murray, W., Saunders, M. A.: Methods for modifying matrix factorizations. Math. Comp. 28, 505–535 (1974)
Michael Gertz, E.: Combination trust-region line-search methods for unconstrained optimization. University of California San Diego 1999
Hallabi, M.El: Globally convergent multi-level inexact hybrid algorithm for equality constrained optimization Département Informatique et Optimisation Institut National des Postes et Télécommunications 1999 Technical Report RT11-98 (revised) Rabat Morocco
Gill, P.E., Hammarling, S.J., Murray, W., Saunders, M.A., Wright, M.H.: Department of Operations Research Stanford University Stanford CA User’s guide for LSSOL (Version 1.0): A Fortran package for constrained linear least-squares and convex quadratic programming Report SOL 86-1 1986
Gill, Philip E., Wright, M.H.: Department of Mathematics University of California San Diego. Course Notes for Numerical Nonlinear Optimization 2001
Moré, Jorge J., Sorensen, D.C.: Computing a Trust Region Step. SIAM J. Sci. Stat. Comput. 4 (3), 553–572 sep(1983)
Nocedal, Jorge, Yuan, Ya Xiang : Combining Trust-Region and Line-Search Techniques. Optimization Technology Center mar OTC 98/04 1998
Powell, M.J.D.: A New Algorithm for Unconstrained Optimization. J. B. Rosen and O. L. Mangasarian and K. Ritter Nonlinear Programming Academic Press 1970
Powell, M.J.D.: A new algorithm for unconstrained optimization A.E.R.E. Harwell 1970 T.P. 382
Powell, M.J.D.: Convergence Properties of a Class of Minimization Algorithms. O. L. Mangasarian and R. R. Meyer and S. M. Robinson Nonlinear Programming 2 Academic Press 1975, pp. 1–27
Stoer, J.: On the numerical solution of constrained least squares problems. SIAM J. Numer. Anal. 8, 382–411 (1971)
Steihaug, T.: The Conjugate Gradient Method and Trust Regions in Large Scale Optimization. SIAM J. Numer. Anal. jun 20 (3), 626–637 1983
Thomas, S.W.: Sequential estimation techniques for quasi-Newton algorithms. Cornell University 1975
Toint, Ph.L.: Towards an efficient sparsity exploiting Newton method for minimization. Sparse Matrices and Their Uses Academic Press New York 1982 I. S. Duff 57–87
Wilson, P.B.: A Simplicial Algorithm for Concave Programming. Harvard University 1963
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Michael Gertz, E. A quasi-Newton trust-region method. Math. Program., Ser. A 100, 447–470 (2004). https://doi.org/10.1007/s10107-004-0511-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-004-0511-1