×

zbMATH — the first resource for mathematics

Conjugate direction algorithms for extended conic functions. (English) Zbl 0597.65059
The author considers the problem of minimizing a function of the form F(q(x),c(x)), where \(q: {\mathbb{R}}^ n\to {\mathbb{R}}\) is a strictly convex twice continuously differentiable quadratic function, \(c: {\mathbb{R}}^ n\to {\mathbb{R}}\) is linear, and \(F: {\mathbb{R}}^ 2\to {\mathbb{R}}\) is twice continuously differentiable and strong monotonically increasing in q. Conjugate direction algorithms are given that minimize functions of this type after a finite number of steps.
Reviewer: M.Bastian
MSC:
65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] J. Abaffy, F. Sloboda: Imperfect conjugate gradient algorithms for extended quadratic functions. Numer. Math. 42 (1983), 97-105. · Zbl 0524.65045 · doi:10.1007/BF01400920 · eudml:132863
[2] P. Bjorstad, J. Nocedal: Analysis of a new algorithm for one-dimensional minimization. Computing 22 (1979), 93-100. · Zbl 0401.65041 · doi:10.1007/BF02246561
[3] C. G. Broyden: Quasi-Newton methods and their application to function minimization. Math. Comp. 21 (1967), 368-381. · Zbl 0155.46704
[4] W. C. Davidon: Variable Metric Method for Minimization. Report ANL-5990 Rev., Argonne National Laboratories, Argonne, IL, 1959.
[5] W. C. Davidon: Conic approximations and collinear scalings for optimizers. SIAM J. Numer. Anal. 17 (1980), 268-281. · Zbl 0424.65026 · doi:10.1137/0717023
[6] E. J. Davison, P. Wong: A robust algorithm that minimizes 1-functions. Automatica 11 (1975), 287-308. · Zbl 0339.65033
[7] L. C. W. Dixon: Conjugate directions without linear searches. Inst. Math. Appl. 11 (1973), 317-328. · Zbl 0259.65060 · doi:10.1093/imamat/11.3.317
[8] L. C. W. Dixon: Conjugate gradient algorithm: quadratic termination properties without line searches. Inst. Math. Appl. 15 (1975), 9-18. · Zbl 0294.90076 · doi:10.1093/imamat/15.1.9
[9] J. Flachs: On the convergence, invariance, and related aspects of a modification of Huang’s algorithm. Optim. Theory Appl. 37 (1982), 315-341. · Zbl 0462.65042 · doi:10.1007/BF00935273
[10] R. Fletcher, M. J. D. Powell. : A rapidly convergent descent method for minimization. Comput. J. 6 (1963), 163-168. · Zbl 0132.11603 · doi:10.1093/comjnl/6.2.163
[11] R. Fletcher, C. M. Reeves: Function minimization by conjugate gradients. Comput. J. 7 (1964), 149-154. · Zbl 0132.11701 · doi:10.1093/comjnl/7.2.149
[12] D. Goldfarb: Extension of Davidon’s variable metric method to maximization under linear inequality and equality constraints. SIAM J. Appl. Math. 17 (1969), 739-763. · Zbl 0185.42602 · doi:10.1137/0117067
[13] M. R. Hestenes, E. Stiefel: The method of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Standards 49 (1952), 409-436. · Zbl 0048.09901
[14] H. Y. Huang: Method of Dual Matrices for Function Minimization. Aero-Astronautic Report No. 88, Rice University, Houston, TX, 1972. · Zbl 0254.65044
[15] M. L. Lenard: Accelerated Conjugate Direction Methods for Unconstrained Optimization. Technical Report No. MRC-1591, University of Wisconsin, Madison, WI, 1976. · Zbl 0352.90058 · doi:10.1007/BF00933252
[16] L. Lukšan: Conjugate gradient algorithms for conic functions. Aplikace matemat. · Zbl 0622.65045
[17] L. Lukšan: Variable metric methods for a class of extended conic functions. Kybernetika 21 (1985), 96-107. · Zbl 0548.90062 · eudml:27959
[18] L. Nazareth: A conjugate direction algorithm without line searches. J. Optim. Theory Appl. 23 (1977), 373-387. · Zbl 0348.65061 · doi:10.1007/BF00933447
[19] J. E. Shirey: Minimization of extended quadratic functions. Numer. Math. 39 (1982), 157-161. · Zbl 0491.65038 · doi:10.1007/BF01408690 · eudml:132787
[20] F. Sloboda: An imperfect conjugate gradient algorithm. Aplikace matematiky 27 (1982), 426-434. · Zbl 0503.65017 · eudml:15263
[21] F. Sloboda: A generalized conjugate gradient algorithm for minimization. Numer. Math. 35 (1980), 223-230. · Zbl 0424.65033 · doi:10.1007/BF01396318 · eudml:186287
[22] D. C. Sorensen: The Q-superlinear convergence of a collinear scaling algorithm for unconstrained optimization. SIAM J. Numer. Anal. 17 (1980), 84-114. · Zbl 0428.65040 · doi:10.1137/0717011
[23] E. Spedicato: A variable-metric method for function minimization derived from invariancy to nonlinear scaling. J. Optim. Theory Appl. 20 (1976), 315-329. · Zbl 0316.90066 · doi:10.1007/BF00933626
[24] G. Zoutendijk: Methods of Feasible Directions. Elsevier, Amsterdam 1960. · Zbl 0097.35408
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.