Abstract
This paper is concerned with collinear scaling algorithms for unconstrained minimization where the underlying local approximants are forced to interpolate the objective function value and gradient at only the two most recent iterates. By suitably modifying the procedure of Sorensen (1980) for deriving such algorithms, we show that two members of the algorithm class derived related to the DFP and BFGS methods respectively are locally and q-superlinearly convergent. This local analysis as well as the results they yield exhibit the same sort of “duality” exhibited by those of Broyden, Dennis and Moré (1973) and Dennis and Moré (1974) for the DFP and BFGS methods. The results in this paper also imply the local and q-superlinear convergence of collinear scaling algorithms of Sorensen (1982, pp. 154–156) related to the DFP and BFGS methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K.A. Ariyawansa and J.G.C. Templeton, “Local and q-superlinear convergence of two collinear scaling algorithms that generalize the DFP and BFGS methods,” Working Paper 83-05, Department of Industrial Engineering, University of Toronto (Toronto, Ont., Canada, 1983).
K.A. Ariyawansa, “Conic approximations and collinear scalings in algorithms for unconstrainted minimization,” Ph.D. Dissertation, Department of Industrial Engineering, University of Toronto (Toronto, Ont., Canada, 1983).
C.G. Broyden, J.E. Dennis Jr. and J.J. Moré, “On the local and superlinear convergence of quasi-Newton methods,”Journal of the Institute of Mathematics and its Applications 12 (1973) 222–236.
W.C. Davidon, “Optimization by non-linear scaling,” in: D. Jacobs, ed.,Proceedings of the Conference on Applications of Numerical Software—Needs and Availability (Academic Press, New York, 1978) pp. 377–383.
W.C. Davidon, “Conic approximations and collinear scalings for optimizers,”SIAM Journal on Numerical Analysis 17 (1980) 268–281.
J.E. Dennis Jr. and J.J. Moré, “A characterization of superlinear convergence and its application to quasi-Newton methods,”Mathematics of Computation 28 (1974) 549–560.
J.E. Dennis Jr. and J.J. Moré, “Quasi-Newton methods, motivation and theory,”SIAM Review 19 (1977) 46–89.
J.E. Dennis Jr. and R.B. Schnabel, “A new derivation of symmetric positive definite secant updates,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming, Vol. 4 (Academic Press, New York, 1980) pp. 167–199.
J.E. Dennis Jr. and H.F. Walker, “Convergence theorems for least-change secant update methods,”SIAM Journal of Numerical Analysis 18 (1981) 949–987.
J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New york, 1970).
D.C. Sorensen, “The q-superlinear convergence of a collinear scaling algorithm for unconstrained optimization,”SIAM Journal on Numerical Analysis 17 (1980) 84–114.
D.C. Sorensen, “Collinear scaling and sequential estimation in sparse optimization algorithms,” in: D.C. Sorensen and R.J.-B. Wets, eds.,Algorithms and Theory in Filtering and Control, Mathematical Programming Study 18 (North-Holland, Amsterdam, 1982) pp. 135–159.
Author information
Authors and Affiliations
Additional information
Research supported in part by funds provided by the Washington State University Research and Arts Committee, by NSF Grant DMS-8414460 and by DOE Grant DE-FG06-85ER25007.
Rights and permissions
About this article
Cite this article
Ariyawansa, K.A. Deriving collinear scaling algorithms as extensions of quasi-Newton methods and the local convergence of DFP- and BFGS-related collinear scaling algorithms. Mathematical Programming 49, 23–48 (1990). https://doi.org/10.1007/BF01588777
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01588777