A class of accelerated conjugate direction methods for linearly constrained minimization problems
HTML articles powered by AMS MathViewer
- by Michael J. Best and Klaus Ritter PDF
- Math. Comp. 30 (1976), 478-504 Request permission
Abstract:
A class of algorithms are described for the minimization of a function of n variables subject to linear inequality constraints. Under weak conditions convergence to a stationary point is demonstrated. The method uses a mixture of conjugate direction constructing and accelerating steps. Any mixture, for example alternation, may be used provided that the subsequence of conjugate direction constructing steps is infinite. The mixture of steps may be specified so that under appropriate assumptions the rate of convergence of the method is two-step superlinear or $(n - p + 1)$-step cubic where p is the number of constraints active at a stationary point. The accelerating step is always superlinearly convergent. A condition is given under which the alternating policy is every step superlinear. Computational results are given for several test problems.References
- Larry Armijo, Minimization of functions having Lipschitz continuous first partial derivatives, Pacific J. Math. 16 (1966), 1–3. MR 191071
- Michael J. Best, A method to accelerate the rate of convergence of a class of optimization algorithms, Math. Programming 9 (1975), no. 2, 139–160. MR 405840, DOI 10.1007/BF01681341
- Michael J. Best and Klaus Ritter, An accelerated conjugate direction method to solve linearly constrained minimization problems, J. Comput. System Sci. 11 (1975), no. 3, 295–322. MR 391501, DOI 10.1016/S0022-0000(75)80055-9
- Jerome Bracken and Garth P. McCormick, Selected applications of nonlinear programming, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233590 A. R. COLVILLE, A Comparative Study on Nonlinear Programming Codes, IBM, New York Scientific Center, Report No. 320-2949, 1968.
- Allen A. Goldstein, Constructive real analysis, Harper & Row, Publishers, New York-London, 1967. MR 0217616
- Klaus Ritter, A superlinearly convergent method for minimization problems with linear inequality constraints, Math. Programming 4 (1973), 44–71. MR 323354, DOI 10.1007/BF01584646
- Klaus Ritter, A method of conjugate directions for linearly constrained nonlinear programming problems, SIAM J. Numer. Anal. 12 (1975), 273–303. MR 383745, DOI 10.1137/0712024
- K. Ritter, Accelerating procedures for methods of conjugate directions, Computing 14 (1975), no. 1-2, 79–105 (English, with German summary). MR 405850, DOI 10.1007/BF02242308
- M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein. MR 0176364 G. ZOUTENDIJK, Methods of Feasible Directions, A Study in Linear and Nonlinear Programming, American Elsevier, New York, 1960. MR 23 #B2156.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Math. Comp. 30 (1976), 478-504
- MSC: Primary 65K05; Secondary 90C30
- DOI: https://doi.org/10.1090/S0025-5718-1976-0431675-3
- MathSciNet review: 0431675