Přehled deterministických a stochastických aproximačních metod pro minimalizaci funkcí

Fabian, Václav

Kybernetika, Tome 1 (1965) no. 6, pp. 499-523

Détail

Classification : 62L20, 65K10

Bibliographie

Ablow C. M., G. Brigham (1955): An analog solution of programming problems. OR 3, 388 to 394. | MR

Abrham J. (1958): Přibližná metoda pro nelinearní programování. Čas. pěst. mat. 83, 425-439. | MR

Abrham J. (1961): An Approximate Method for Convex Programming. Econometrica 29, 700 to 703. | MR

Abrham J. (1962): The Multiplex Method and its Application to Concave Programming. CZM 12, 325-345. | MR

Ackoff R. L. (1961): Progress in Operations Research. Vol. I. John Wiley & Sons. Inc., New York - London. | MR

Akaike H. (1959): On successive transformation of probability distribution and its application to the analysis of the optimum gradient method. Ann. Inst. Statist. Math. Tokyo 11, 1 - 16. | MR

Altman M. (1964): Stationary points in non-linear programming. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 12, 29-35. | MR

Archer D. H. (1960): An optimising control for the chemical process industries. Britisch Chemical Engineering, 88 -94.

Arnoff E. L., S. S. Sengupta (1961): Mathematical Programming. Viz sb. Ackoff (1961), 105-210. | MR

Arrow K. J., L. Hurwicz (1951): A gradient method for approximation of saddle points and conditionned maxima. Rep. 253 RAND Corporation, June 13.

Arrow K. J., L. Hurwicz (1956): Reduction of constrained maxima to saddle-point problems. Proc. 3rd Berkeley Symp. Math., Statist. Probability, vol. V, 1 - 20, Un. Calif. Press. | MR

Arrow K. J., L. Hurwicz H. Uzawa (1958): Studies in Linear and Non-linear Programming. Stanford University Press, Stanford, California. | MR

Beale E. M. L. (1955): On optimising a convex function subject to linear inequalities. JRSS B 17, 173-184. | MR

Beale E. M. L. (1958): On an iterative method for finding a local minimum of a function of more than one variable. Statist. Tech. Research Group Techn. Report No. 25, Princeton Univ., Princeton, N. Y.

Beale E. M. L. (1959): On quadratic programming. Naval Research Logistic Quart. 6, 227 - 243. | MR

Bellman R. (1957): Dynamic Programming. Princeton University Press, Princeton, N. Y. | MR

Block H. D. (1957): On stochastic approximation. ONR Report, Dept. of Mathematics. Cornell University, Ithaca, New York, 13 pp.

Blum J. R. (1954a): Approximation methods which converge with probability one. AMS 25, 382 to 386. | MR

Blum J. R. (1954b): Multidimensional stochastic approximation methods. AMS 25, 463 - 483. | MR

Blum J. R. (1958): A note on stochastic approximation. Proc. Am. Math. Soc., 9, 404-407. | MR

Both A. D. (1949): An application of the method of steepest descent to the solution of systems of non-linear simultaneous equations. Quar. J. Mech. Appl. Math. 2, 460 -468. | MR

Box G. E. P., K. B. Wilson (1951): On the experimental attainment of optimum conditions. JRSS B 13, 1-45. | MR

Box G. E. P. (1954): The exploration and exploitation of response surfaces: some general considerations and examples. Biometrics 10, 16-60.

Box G. E. P., P. V. Youle (1955): The exploration and exploitation of response surfaces: an example of the link between the fitted surface and the basic mechanism of the system. Biometrics 11, 287-323.

Box G. E. P. (1957): Evolutionary operation: a method for increasing industrial productivity. Applied Statistics 6, 81 - 101.

Box G. E. P., J. S. Hunter (1959): Condensed calculations for evolutionary operation programs. Technometrics, 77-95. | MR

Brooks S. H. (1959): A comparison of maximum seeking methods. J. Oper. Research Soc. 7, 430-457.

Brownlee K. A. J. L. Hodges, Jr., M. Rosenblatt (1963): The Up-and-Down method with small samples. JASA 48, 262-277. | MR

Burkholder D. L. (1956): On a certain class of stochastic approximation procedures. AMS 27, 1044-1059. | MR

Carroll Ch. W. (1961): The created response surface technique for non-linear restrained systems. OR 9, 169-185. | MR

Cauchy A. (1847): Méthode générale pour la résolution des systèms d'equations simultanées. Compte Rendu, Ac. Sci. Paris, 25, 536-538.

Cheney E. W., A. A. Goldstein (1959): Newton's method of convex programming and Tchebycheff approximation. Numerische Math. 1, 253 - 268. | MR

Chung K. L. (1954): On a stochastic approximation method. AMS 25, 463 - 483. | MR

Crockett J. B., H. Chernoff (1955): Gradient methods of maximization. Pacif. J. Math. 5, 33-50. | MR

Curry H. B. (1944): The method of steepest descent for non-linear minimization problems. Quarterly Appl. Math. 2, 258-261. | MR

DeLand E. C. (1959): Continuous programming methods on an analog computer. Rand Corpor. Paper No. P-1815, Sept.

Derman C. (1956): An application of Chung's lemma to the Kiefer-Wolfowitz stochastic approximation procedure. AMS 27, 532-536. | MR

Derman C. (1956): Stochastic approximation. AMS 27, 879-886. | MR

Derman C. (1957): Non-parametric Up-and-Down experimentations. AMS 28, 795 - 798. | MR

Derman C., J. Sacks (1959): On Dvoretzky's stochastic approximation theorem. AMS 30, 601-606. | MR

Dixon W. J., A. M. Mood (1948): A method for obtaining and analyzing sensitivity data. JASA 43, 109-126.

Driml M., O Hanš (1960): On experience theory problems. PRTR2, 93 - 111. | MR

Driml M., O. Hanš (1960): Continuous stochastic approximation. PRTR2, 113-122. | MR

Driml M., J. Nedoma (1960): Stochastic approximation for continuous random processes. PRTR2, 145-158. | MR

Dupač V. (1957): O Kiefer-Wolfowitzově aproximační metodě. Čas. pěst. mat. 82, 47 - 75. | MR

Dupač V. (1958): Notes on stochastic approximation methods. CZM 8 (83), 139-149. | MR

Dupač V. (1965a): A dynamic stochastic approximation method. V tisku v AMS 36. | MR

Dupač V. (1965b): A dynamical stochastic approximation method. Referát na 4. pražské konferenci o teorii informace, statistických rozhodovacích funkcích a náhodných procesech.

Dvoretzky A. (1956): On stochastic approximation. Proc. Third. Berkeley Symp. Math. Statistics and Probability, Vol. I, 39-55. University of California Press. | MR

Fabian V. (1960): Stochastic approximation methods. CZM 10 (85), 123-159. | MR

Fabian V. (1961): Stochastická aproximační metoda pro hledání optimálních podmínek v experimentální práci a v samoadaptivních systémech. Aplikace matematiky, 6, 162-183. | MR

Fabian V. (1962): Blokové logické schema automatického optimalizátoru. Aplikace matematiky 7, 426-440.

Fabian V. (1964): A new one-dimensional stochastic approximation method for finding a local minimum of a function. PRTR3, 85-105. | MR

Fabian V. (1965): Stochastic approximation of constrained minima. Referát na 4. pražské konferenci o teorii informace, statistických rozhodovacích funkcích a náhodných procesech.

Fabius J. (1959): Stochastic approximation. Statistics Neerlandica 13, 445-452. | MR

Фаддеев Д. К. B. H. Фаддеевa (1963): Вычислительные методы линейной алгебры. Москва.

Farkas J. (1902): Über die Theorie der einfachen Ungleichungen. Zeit. Reine Angew. Math. 124, 1-24.

Farrell R. H. (1962): Bounded length confidence intervals for the zero of a regression funkction. AMS 33, 237-247. | MR

Фельдбаум А. А. (1958): Автоматический оптимизатор. Автоматика и телемeхaника 14, 731-743.

Fiacco A. V., G. P. McCormick (1964a): The sequential unconstrained minimization technique for nonlinear programming, a primal-dual method. Management Science 10, 360 - 366. | MR

Fiacco A. V., G. P. McCormick (1964b): Computational algorithm for the sequential unconstrained minimization technique for nonlinear programming. Management Science 10, 601-617.

Finkel R. W. (1959): The method of resultant descents for the minimization of an arbitrary function. Space Technology Lab. Tech. Report, Los Angeles.

Fletcher R., M. J. D. Powell (1963/4): A rapidly convergent descent method for minimization. Comp. J. 6, 163 - 168. | MR

Forsythe A. I., G. E. Forsythe (1952): Punched-cards experiments with accelerated gradient method for linear equations. NBS Report 1643, National Bureau of Standards, Los Angeles.

Forsythe G. E., T. S. Motzkin (1951): Assymptotic properties of the optimum gradient method. Bull. Amer. Math. Soc. 57, 73.

Forsythe G. E. (1953): Solving linear algebraic expressions can be interesting. Bull. Amer. Math. Soc. 59, 299-329. | MR

Forsythe G. E., W. R. Wasow (1959): Finite-difference methods for partial difference equations. New York, Wiley & Sons.

Фридман B. M. (1962): О сходимости методов типа найскорейшего спуска. Усп. мат. наук 17, 201-204. | MR

Friedman M., L. J. Savage (1947): Planning experiments seeking maxima. Select. Techn. of Statist. Analysis. McGraw-Hill, New York.

Friedman S.. (1963): On stochastic approximations. AMS 34, 343 - 346. | MR

Frish R. (1957): The multiplex method for linear programming. Sankhya 18, 329-362. | MR

Frish R. (1958): The multiplex method for linear programming. Memorandum fra Socialekonomisk Institut, Universitetet Oslo 12.

Gardner L. A., Jr. (1963): Stochastic approximation and its application to problems of prediction and control synthesis. Internat. Symp. Nonlinear Differential Equations and Nonlinear Mech., Colorado Springs, New York -London, Acad. Press, 241 - 258. | MR

Gardner L. A., Jr. (1964): Adaptive predictors. PRTR3. | MR

Germansky B. (1934): Notiz über die Lösung von Extemaufgaben mittels Iteration. Z. Ang. Math. Mech. 14, 187-187.

Гладышев Е. Г.: O стохастической аппроксимации. Теория вероятностей и прим. 10, 297-300. | Zbl

Glass H., L. Cooper (1965): Sequential search: a method for solving constrained optimization problems. JACM 12, 71-82.

Goldstein A. A. (1962): Cauchy's method of minimization. Numer. Math. 4, 146-150. | MR

Goldstein A. A., Ward Cheney (1958): A finite algorithm for the solutions of consistent linear equations and inequalities and for the Tchebycheff approximation of inconsistent linear equations. Pacific J. Math. 8, 415-427. | MR

Griffith R. E., R. A. Stewart (1961): A nonlinear programming technique for the optimization of continuous processing systems. Management Science 7, 379-392. | MR

Guttman L., R. Guttman (1959): An illustration of the use of stochastic approximation methods. Biometrics 15, 551 - 559.

Hanš O., A. Špaček (1960): Random fixed point approximation by differentiable trajectories. PRTR2, 203-213. | MR

Hartley H. O., R. R. Hocking (1963): Convex programming by tangential approximation. Management Science 9, 600-612. | MR

Hestens M. R. (1951): Iterative methods for solving linear equations. NAML Report 52-9, Nat. Bureau of Standards, Los Angeles, 19 pp.

Ho Yu Chi (1962): On the stochastic approximation method and optimal filtering theory. Journal of math. analysis and applications 6, 152-154. | MR

Hodges J. L., Jr., E. L. Lehmann (1956): Two approximation to the Robbins Monro process. Proc. 3rd Berkeley Symp. on Math. Statistics and Prob., vol. 1, 95-104. | MR

Hooke R., T. A. Jeeves (1961): Direct search solution of numerical and statistical problems. JACM 8, 212-

Hotelling H. (1941): Experimental determination of the maximum of a function. AMS 12, 20-46. | MR

Householder A. S., F. L. Bauer (1960): Certain iterative methods for solving linear systems. Numerische Mathematik 2, 55-59. | MR

Иванов B. B. (1962): Oб алгоритмах быстрого спуска. ДAH CCCP 143, 775-778. | MR

John F. (1964): Extremum problems with inequalities as subsidiary conditions. „Studies and Essays", Courant anniversary volume, pp. 187-204. | MR

Kantorovič L. V. (1945): On an effective method of solving extremal problems for quadratic functional. C. R. de l'Ac. Sci. de l'URSS XLVIII, no. 7, 455-459. | MR

Канторович Л. B. (1945): 0б oднom методе решения экстремальных задач для квадратных функциохалов. ДАН CCCP 48, 483-487.

Канторович Л. B. (1947): О методе найскорейшего спуска. ДAH CCCP 56, 233-236 | MR

Канторович Л. B. (1948): Функциональный анализ и прокладная математика. Усп. мат. наук 3, 89-185. | MR

Kelley J. E. (1960): The cutting-plane method for solving convex programs. JSIAM 8, 703-712. | MR

Kerner M. (1933): Die Differentiale in der allegemeinen Analysis. Ann. of Math. 34, 546- 572. | MR

Kerstukos A., R. I. van Nice (1958): Optimizing process operations. Automation 5.

Kesten H. (1958): Accelerated stochastic approximation. AMS 29, 41-59. | MR

Kiefer J., J. Wolfowitz (1952): Stochastic estimation of the maximum of a regression function. AMS 23, 462-466. | MR

Kiefer J. (1953): Sequential minimax search for a maximum. Proc. Amer. Math. Soc. 4, 502-506. | MR

Kiefer J. (1957): Optimum sequential search and approximation methods under minimum regularity assumptions. JSIAM 5, 105-136. | MR

Killianpur G. (1954): A note on the Robbins-Monro stochastic approximation method. AMS 25. | MR

Kitagawa T. (1959): Successive processes of statistical controls (2). Met. Coll. Sci. Univ. Kyoto, Ser. A, Math. 13, 1-16. | MR

Kizner W. (1964): A numerical method for finding solutions of nonlinear equations. JSIAM 12, 424-428. | MR

Klingman W. R., D. M. Himmelblau (1964): Nonlinear programming with the aid of a multiplegradient summation technique. JACM 11, 400-415. | MR

Kose T. (1956): Solutions of saddle value problems by differential equations. Econometrica 24, 59-70. | MR

Красулина Т. П. (1962): 3амечания о некотопых процесах стохастической апроксимации. Теория вероят к прим. 7, 113-118.

Krolak P., L. Cooper (1963): An extension of Fibonaccian search to several variables. CACM 6, 639-641. | MR

Kuhn H. W., A. W. Tucker (1950): Nonlinear programming. Proc. 2nd Berkeley Symp. on Mathem. Statist. and Probability, Univ. of Calif. Pres., 481-492. | MR

Lanczos C. (1952): Solution of systems of linear equations by minimized iterations. J. Res. Nat. Bureau of Standards 49, 33-53. | MR

Levenberg K. (1944): A method for the solution of certain non-linear problems in least squares. Qu. Appl. Math. 2, 164-168. | MR

Leżański T. (1963): Über das Minimumproblem für Funktionale in Banachschen Räumen. Math. Annalen 152, 271-274. | MR

Maňas M. (1965): Nelineární programování. Ek. mat. obzor 1, 3-34.

Mangasarian O. I. (1962): Duality in nonlinear programming. Qu. Appl. Math. 20, 300-302. | MR

Mangasarian O. I. (1963): Equivalence in nonlinear programming. Naval. Res. Logist. Quart. 10, 299-306. | MR

Manne A. S. (1953): Concave programming for gasoline blends. Rand. Corp. Paper. No. P-383.

Marquardt D. W. (1963): An algorithm for least-squares estimation of nonlinear parameters. JSIAM 11, 431-441. | MR

Mises R. von, H. Pollaczek-Geiringer (1929): Praktische Verfahren der Gleichungsauflösung. Z. Angew. Math. Mech. 9, 58-77.

Nashed M. Z. (1964): The convergence of the method of steepest descent for nonlinear equations with variational or quasi-variational operators. J. Math. Mech. 13, 765-794. | MR

Nashed M. Z. (1965): On general iterative methods of the solutions of a class of nonlinear operator equations. Math. of Computation 19, 14-24. | MR

Nelder J. A., R. Mead (1965): A simplex method for function minimization. Comp. J. 7, 308-313.

Newman D. J. (1960): Locating the maximum on a unimodal surface. Presented at the 18th National Meeting of Operations Res. Soc., October 1960, neuveřejněno, viz Spang (1962).

Newman D. J. (1965): Location of the maximum on unimodal surfaces. JACM 12, 395-398. | MR

Pietrzykowski T. (1961): On an iteration method for maximizing a concave function on a convex set. Práce ZAM, A 13. | MR

Pietrzykowski T. (1962): On a method of approximative finding conditional maximums. Algorytmy 1, 9-15. | MR

Powell M. J. D. (1964): An efficient method for finding the mininum of a function of several variables without calculating the derivatives. Computer Journal 7, 155-162. | MR

Powell M. J. D. (1965): A method for minimizing a sum of squares of non-linear functions without calculating derivatives. Comp. J. 7, 303 - 307. | MR

Reeves C. M. (1964): Algorithm 238, conjugate gradient method. CACM 7, 481.

Remez E. (1934): Sur un procédé convergent d'approximations succesives pour determiner les polynomes d'approximation. C. R. Ac. Sci. Paris 198, 2063-2065.

Robbins H. S. Monro (1951): A stochastic approximation method. AMS 22, 400-407. | MR

Rosen J. B. (1960): The gradient projection method for nonlinear programming. I. Linear constrainsts. JSIAM 8, 181-217. | MR

Rosen J. B. (1961): The gradient projection method for nonlinear programming. Part. II. Nonlinear constraints. JSIAM 9, 514-532. | MR

Rosenbloom P. C. (1956): The method of steepest descent. Proc. of Symp. in Applied Math. vol. 6, pp. 127-176. | MR

Rosenbrock H. H. (1960): An automatic method for finding the largest or least value of a function. Computer Journal 3, 175-184. | MR

Saaty T. L., Bram J. (1964): Nonlinear mathematics. McGraw-Hill Book Co., New York. | MR

Sacks J. (1958): Asymptotic distributions of stochastic approximatons. AMS 29, 373-405. | MR

Sakrison D. J. (1962a): Design of filters for non-mean-square error performance criteria by means of a continuous adjustment procedure. Quarterly Progress Report No. 66, 189-201 and No. 67, 119-126, Research Lab. of Electronics.

Sakrison D. J. (1962b): Application of stochastic approximation methods to system optimization. Techn. Rep. 391, July 10, Massachusetts Inst. Technology, Res. Lab. Electronics. Cambridge, Massachusetts.

Sakrison D. J. (1963): Iterative design of optimum filters for non-mean-square error performance criteria. Trans. of the IEEE Professional Group on Information Theory 9, 161-167.

Sakrison D. J. (1964): A continuous Kiefer-Wolfowitz procedure for random processes. AMS 35, 590-599. | MR

Schmetterer L. (1953): Bemerkungen zum Verfahren der stochastischen Iteration. Österreichisches Ingenieur-Archiv VII, 111-117. | MR

Schmetterer L. (1954a): Sur l'approximation stochastique. Bull. Inst. Internat. Statist. 24. 203 to 206. | MR

Schmetterer L. (1954b): Zum Sequentialverfahren von Robbins und Monro. Monatshefte für Mathematik 58, 33-37. | MR

Schmetterer L. (1958): Sur l'iteration stochastique. Colloques Internationaux du centre national de la recherch scientifique. 87, 55-63. | MR

Schmetterer L. (1961): Stochastic approximation. Proc. 4th Berkeley Sympos. Math. Statist. Probab. 1, 587-609. | MR

Schröder E. (1870): Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317-365. | MR

Shah B. V., Buehler R. J., Kempthorne O. (1964): Some algorithms for minimizing a function of several variables. JSIAM 12, 74-92. | MR

Spang H. A. III (1962): A review of minimization techniques for nonlinear functions. SIAM Review 4, 343-365. | MR

Стаховский Р. И. (1958): Двухканальный автоматический оптимизатор. Автоматика и телемеханика, 14, 744-756.

Stiefel E. (1952): Über einige Methoden der Relaxationsrechnung. Zeit. Angew. Math. Phys. 3, 1-33. | MR

Stiefel E. (1955): Relaxation Methoden bester Strategie zur Lösung linearer Gleichungssysteme. Comm. Math. Helv. 29, 157-159. | MR

Ваинбепг М. М. (1960): О сходимости метода найскорейшего спуска для нелнейных уравнений. ДАН СССР, 130, 9-12.

Vosiková M. (1964): Asymptotické vlastnosti Kiefer-Wolfowitzovy aproximační metody. Dipl. práce, Mat.-fyz. fak. KU.

Wells M. (1965): Algorithm 251, Function minimization [E 4]. CACM 8, 169-170.

Wetherill G. B. (1963): Sequential estimation of quantal response curves (with discussion). JRS. B 25, 1-48.

Witzgall C. (1960): Gradient-projection methods for linear programming. Princeton Un. and IBM Corp. Rep. No. 2,

Wolfe P. (1960): Accelerating the cutting-plane method for nonlinear programming. Rand. Corp. Paper No. P-2010. | MR

Wolfe P. (1961): Accelerating the cutting-plane method for nonlinear programming. JSIAM 9, 481-488. | MR

Wolfe P. (1962): Recent developments in nonlinear programming. Advance in computers, vol. 3, Acad. Press. New York, 155-187. | MR

Wolfowitz (1952): On the stochastic approximation method of Robbins and Monro. AMS 23, 457-561. | MR

Wolfowitz (1956): On stochastic approximation methods. AMS 27, 1151 - 1155. | MR

Zoutendijk G. (1959): Maximizing a function in a convex region. JRSS B 21, 338 - 355. | MR

Zoutendijk G. (1960): Methods of feasible directions. Elsevier Publ. Co., Amsterdam-London -New York -Princeton. | MR