The theory of dynamic programming
Author:
Richard Bellman
Journal:
Bull. Amer. Math. Soc. 60 (1954), 503515
MathSciNet review:
0067459
Fulltext PDF
References 
Additional Information
 1.
K.
J. Arrow, D.
Blackwell, and M.
A. Girshick, Bayes and minimax solutions of sequential decision
problems, Econometrica 17 (1949), 213–244. MR 0032173
(11,261h)
 2.
K. J. Arrow, T. E. Harris, and J. Marschak, Optimal inventory policy, Cowles Commission Paper No. 44, 1951.
 3.
Richard
Bellman, An introduction to the theory of dynamic programming,
The Rand Corporation, Santa Monica, Calif., 1953. MR 0061805
(15,887e)
 4.
Richard
Bellman, On games involving bluffing, Rend. Circ. Mat. Palermo
(2) 1 (1952), 139–156. MR 0053470
(14,778h)
 5.
Richard
Bellman, On the theory of dynamic programming, Proc. Nat.
Acad. Sci. U. S. A. 38 (1952), 716–719. MR 0050856
(14,392b)
 6.
Richard
Bellman, Some problems in the theory of dynamic programming,
Econometrica 22 (1954), 37–48. MR 0060709
(15,713c)
 7.
Richard
Bellman, Bottleneck problems and dynamic programming, Proc.
Nat. Acad. Sci. U. S. A. 39 (1953), 947–951. MR 0061808
(15,888a)
 8.
Richard
Bellman, An introduction to the theory of dynamic programming,
The Rand Corporation, Santa Monica, Calif., 1953. MR 0061805
(15,887e)
 9.
Richard
Bellman, Some functional equations in the theory of dynamic
programming, Proc. Nat. Acad. Sci. U. S. A. 39
(1953), 1077–1082. MR 0061806
(15,887f)
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Richard
Bellman, Dynamic programming and a new formalism in the calculus of
variations, Proc. Nat. Acad. Sci. U. S. A. 40 (1954),
231–235. MR 0061289
(15,804f)
 11.
R. Bellman, The theory of dynamic programming, a general survey, Chapter from "Mathematics for Modern Engineers" by E. F. Beckenbach, McGrawHill, forthcoming.
 12.
Richard
Bellman, On some applications of the theory of dynamic programming
to logistics, Naval Res. Logist. Quart. 1 (1954),
141–153. MR 0063636
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Richard
Bellman, Some applications of the theory of dynamic
programming—a review, J. Operations Res. Soc. Amer.
2 (1954), 275–288. MR 0062414
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Richard
Bellman, Bottleneck problems, functional equations, and dynamic
programming, Econometrica 23 (1955), 73–87. MR 0070935
(17,58e)
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R. Bellman, On a functional equation arising in the problem of optimal inventory, The RAND Corporation, Paper P480, January 1954.
 16.
R. Bellman, Dynamic programming and the calculus of variationsI, The RAND Corporation, Paper P495, March 1954.
 17.
Richard
Bellman, Dynamic programming of continuous processes, The Rand
Corporation, Santa Monica, Calif., 1954. MR 0071700
(17,171a)
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Richard
Bellman, Dynamic programming of continuous processes, The Rand
Corporation, Santa Monica, Calif., 1954. MR 0071700
(17,171a)
 19.
Richard
Bellman and David
Blackwell, Some twoperson games involving bluffing, Proc.
Nat. Acad. Sci. U. S. A. 35 (1949), 600–605. MR 0031700
(11,192b)
 20.
Richard
Bellman, Irving
Glicksberg, and Oliver
Gross, On some variational problems occurring in the theory of
dynamic programming, Proc. Nat. Acad. Sci. U. S. A.
39 (1953), 298–301. MR 0054183
(14,885d)
 21.
R. Bellman, I. Glicksberg, and O. Gross, On some variational problems in the theory of dynamic programming, Rend. Circ. Mat. Palermo, forthcoming.
 22.
R.
Bellman, I.
Glicksberg, and O.
Gross, The theory of dynamic programming as applied to a smoothing
problem, J. Soc. Indust. Appl. Math. 2 (1954),
82–88. MR
0067378 (16,721c)
 23.
Richard
Bellman and Oliver
Gross, Some combinatorial problems arising in the theory of
multistage processes, J. Soc. Indust. Appl. Math. 2
(1954), 175–183 (1955). MR 0068193
(16,843a)
 24.
R. Bellman, T. E. Harris, and H. N. Shapiro, Studies on functional equations occurring in decision processes, The RAND Corporation, Paper P382, August 1952.
 25.
Richard
Bellman and Shermann
Lehman, On the continuous goldmining equation, Proc. Nat.
Acad. Sci. U. S. A. 40 (1954), 115–119. MR 0060681
(15,708c)
 26.
R. Bellman and R. S. Lehman, On a functional equation in the theory of dynamic programming and its generalizations, The RAND Corporation, Paper P433, January 1954.
 27.
R. Bellman and R. S. Lehman, Studies on bottleneck problems in production processes, The RAND Corporation, Paper P492, February 1954.
 28.
R. R. Bush and C. F. Mosteller, A mathematical model for simple learning, Psychological Review vol. 58 (1951) pp. 313325.
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A.
Dvoretzky, J.
Kiefer, and J.
Wolfowitz, The inventory problem. I. Case of known distributions of
demand, Econometrica 20 (1952), 187–222. MR 0047304
(13,856a)
 30.
A.
Dvoretzky, A.
Wald, and J.
Wolfowitz, Elimination of randomization in certain statistical
decision procedures and zerosum twoperson games, Ann. Math.
Statistics 22 (1951), 1–21. MR 0039228
(12,515a)
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W. K. Estes, Toward a statistical theory of learning, Psychological Review vol. 57 (1950) pp. 94107.
 32.
M. M. Flood, On stochastic learning theory, The RAND Corporation, Paper P353, December 1952.
 33.
S. Johnson, Optimal two and threestage production schedules with setup times included, The RAND Corporation, Paper P402, May 1953.
 34.
S. Johnson and S. Karlin, On optimal sampling procedure for a problem of two populationsI, The RAND Corporation, Paper P328, October 1952.
 35.
S. Karlin, A mathematical treatment of learning modelsI, The RAND Corporation, Research Memorandum RM921, September 1952.
 36.
S. Karlin, and H. N. Shapiro, Decision processes and functional equations, The RAND Corporation, Research Memorandum RM933, September 1952.
 37.
M. Peisakoff, More on games of survival, The RAND Corporation, Research Memorandum RM884, June 1952.
 38.
Herbert
Robbins, Some aspects of the sequential design
of experiments, Bull. Amer. Math. Soc. 58 (1952), 527–535.
MR
0050246 (14,300b), http://dx.doi.org/10.1090/S000299041952096208
 39.
L.
S. Shapley, Stochastic games, Proc. Nat. Acad. Sci. U. S. A.
39 (1953), 1095–1100. MR 0061807
(15,887g)
 40.
Richard
Bellman, Dynamic programming of continuous processes, The Rand
Corporation, Santa Monica, Calif., 1954. MR 0071700
(17,171a)
 41.
Richard
Bellman, Some functional equations in the theory of dynamic
programming, Proc. Nat. Acad. Sci. U. S. A. 39
(1953), 1077–1082. MR 0061806
(15,887f)
 42.
R. Bellman, An iterative procedure for the determination of the Perron root of a positive matrix, The RAND Corporation, Paper P577.
 43.
R. Bellman, On a quasilinear equation, The RAND Corporation, Paper P575.
 44.
Richard
Bellman, Dynamic programming and a new formalism in the theory of
integral equations, Proc. Nat. Acad. Sci. U.S.A. 41
(1955), 31–34. MR 0072365
(17,272d)
 45.
R. Bellman, A problem in the sequential design of experiments, The RAND Corporation, Paper P586.
 46.
Richard
Bellman, Decision making in the face of uncertainty. I, Naval
Res. Logist. Quart. 1 (1954), 230–232 (1955). MR 0067446
(16,730g)
 47.
Richard
Bellman, Decision making in the face of uncertainty. I, Naval
Res. Logist. Quart. 1 (1954), 230–232 (1955). MR 0067446
(16,730g)
 48.
Richard
Bellman, Some problems in the theory of dynamic programming,
Econometrica 22 (1954), 37–48. MR 0060709
(15,713c)
 49.
R. Bellman, I. Glicksberg, and O. Gross, On the optimal inventory equation, The RAND Corporation, Paper P572.
 50.
R. Bellman, On some mathematical problems arising in the theory of optimal inventory and stock control, The RAND Corporation, Paper P580.
 1.
 K. J. Arrow, D. Blackwell, and M. A. Girshick, Bayes and minimax solutions of sequential decision problems, Econometrica vol. 17 (1949) pp. 214244. MR 32173
 2.
 K. J. Arrow, T. E. Harris, and J. Marschak, Optimal inventory policy, Cowles Commission Paper No. 44, 1951.
 3.
 R. Bellman, An introduction to the theory of dynamic programming, The RAND Corporation, Report R245, 1953. MR 61805
 4.
 R. Bellman, On games involving bluffing, Rend. Circ. Mat. Palermo (2) vol. 1 (1952) pp.118. MR 53470
 5.
 R. Bellman, On the theory of dynamic programming, Proc. Nat. Acad. Sci. U.S.A. vol. 38 (1952) pp. 716719. MR 50856
 6.
 R. Bellman, Some problems in the theory of dynamic programming, Econometrica vol. 22 (1954) pp. 3748. MR 60709
 7.
 R. Bellman, On bottleneck problems and dynamic programming, Proc. Nat. Acad. Sci. U.S.A. vol. 39 (1953) pp. 947951. MR 61808
 8.
 R. Bellman, On computational problems in the theory of dynamic programming, Symposium of Numerical Methods, Santa Monica, 1953, The RAND Corporation, Paper P423. MR 61805
 9.
 R. Bellman, Some functional equations in the theory of dynamic programming, Proc. Nat. Acad. Sci. U.S.A. vol. 39 (1953) pp. 10771082. MR 61806
 10.
 R. Bellman, Dynamic programming and a new formalism in the calculus of variations, Proc. Nat. Acad. Sci. U.S.A. vol 40 (1954) pp. 231235. MR 61289
 11.
 R. Bellman, The theory of dynamic programming, a general survey, Chapter from "Mathematics for Modern Engineers" by E. F. Beckenbach, McGrawHill, forthcoming.
 12.
 R. Bellman, Some applications of the theory of dynamic programming to logistics, Navy Quarterly of Logistics, September 1954. MR 63636
 13.
 R. Bellman, Some applications of the theory of dynamic programminga review, Operations Research Quarterly, June 1954. MR 62414
 14.
 R. Bellman, Bottleneck problems, functional equations, and dynamic programming, The RAND Corporation, Paper P483, January 1954; Econometrica (to appear). MR 70935
 15.
 R. Bellman, On a functional equation arising in the problem of optimal inventory, The RAND Corporation, Paper P480, January 1954.
 16.
 R. Bellman, Dynamic programming and the calculus of variationsI, The RAND Corporation, Paper P495, March 1954.
 17.
 R. Bellman, Dynamic programming and the calculus of variationsII, The RAND Corporation, Paper P512, April 1954. MR 71700
 18.
 R. Bellman, Monotone convergence in dynamic programming and the calculus of variations, The RAND Corporation, Paper P513, April 1954. MR 71700
 19.
 R. Bellman and D. Blackwell, Some twoperson games involving bluffing, Proc. Nat. Acad. Sci. U.S.A. vol. 35 (1949) pp. 600605. MR 31700
 20.
 R. Bellman, I. Glicksberg, and O. Gross, On some variational problems occurring in the theory of dynamic programming, Proc. Nat. Acad. Sci. U.S.A. vol. 39 (1953) pp. 298301. MR 54183
 21.
 R. Bellman, I. Glicksberg, and O. Gross, On some variational problems in the theory of dynamic programming, Rend. Circ. Mat. Palermo, forthcoming.
 22.
 R. Bellman, I. Glicksberg, and O. Gross, The theory of dynamic programming as applied to a smoothing problem, Journal of the Society for Industrial and Applied Mathematics, forthcoming. MR 67378
 23.
 R. Bellman and O. Gross, Some combinatorial problems arising in the theory of multistage processes, The RAND Corporation, Paper P456, November 1953. MR 68193
 24.
 R. Bellman, T. E. Harris, and H. N. Shapiro, Studies on functional equations occurring in decision processes, The RAND Corporation, Paper P382, August 1952.
 25.
 R. Bellman and R. S. Lehman, On the continuous goldmining equation, Proc. Nat. Acad. Sci. U.S.A. vol. 40 (1954) pp. 115119. MR 60681
 26.
 R. Bellman and R. S. Lehman, On a functional equation in the theory of dynamic programming and its generalizations, The RAND Corporation, Paper P433, January 1954.
 27.
 R. Bellman and R. S. Lehman, Studies on bottleneck problems in production processes, The RAND Corporation, Paper P492, February 1954.
 28.
 R. R. Bush and C. F. Mosteller, A mathematical model for simple learning, Psychological Review vol. 58 (1951) pp. 313325.
 29.
 A. J. Dvoretzky, J. Kiefer, and J. Wolfowitz, The inventory problemI: Case of known distributions of demand, and The inventory problemII: Case of unknown distributions of demand, Econometrica vol. 20 (1952) pp. 187222. MR 47304
 30.
 A. J. Dvoretzky, A. Wald, and J. Wolfowitz, Elimination of randomization in certain statistical decision procedures and zerosum twoperson games, Ann. Math. Statist, vol. 22 (1951) pp. 121. MR 39228
 31.
 W. K. Estes, Toward a statistical theory of learning, Psychological Review vol. 57 (1950) pp. 94107.
 32.
 M. M. Flood, On stochastic learning theory, The RAND Corporation, Paper P353, December 1952.
 33.
 S. Johnson, Optimal two and threestage production schedules with setup times included, The RAND Corporation, Paper P402, May 1953.
 34.
 S. Johnson and S. Karlin, On optimal sampling procedure for a problem of two populationsI, The RAND Corporation, Paper P328, October 1952.
 35.
 S. Karlin, A mathematical treatment of learning modelsI, The RAND Corporation, Research Memorandum RM921, September 1952.
 36.
 S. Karlin, and H. N. Shapiro, Decision processes and functional equations, The RAND Corporation, Research Memorandum RM933, September 1952.
 37.
 M. Peisakoff, More on games of survival, The RAND Corporation, Research Memorandum RM884, June 1952.
 38.
 H. Robbins, Some aspects of the sequential design of experiments, Bull. Amer. Math. Soc. vol. 58 (1952) pp. 527536. MR 50246
 39.
 L. Shapley, Stochastic games, Proc. Nat. Acad. Sci. U.S.A. vol. 39 (1953) pp. 10951100. MR 61807
 40.
 R. Bellman, Dynamic programming and continuous processes, The RAND Corporation, Report R271, November 1954. MR 71700
 41.
 R. Bellman, Some functional equations in the theory of dynamic programmingI, Point functions and point transformations, The RAND Corporation, Paper P566, September 1954. MR 61806
 42.
 R. Bellman, An iterative procedure for the determination of the Perron root of a positive matrix, The RAND Corporation, Paper P577.
 43.
 R. Bellman, On a quasilinear equation, The RAND Corporation, Paper P575.
 44.
 R. Bellman, Dynamic programming and a new formalism in the theory of integral equations, Proc. Nat. Acad. Sci. U.S.A., to appear. MR 72365
 45.
 R. Bellman, A problem in the sequential design of experiments, The RAND Corporation, Paper P586.
 46.
 R. Bellman, Decisionmaking in the face of uncertaintyI, Navy Quarterly of Logistics, September 1954. MR 67446
 47.
 R. Bellman, Decisionmaking in the face of uncertaintyII, Navy Quarterly of Logistics, December 1954. MR 67446
 48.
 R. Bellman, Dynamic programming and multistage decision processes of stochastic type, Symposium on Linear Programming, Washington, D. C., December, 1954. MR 60709
 49.
 R. Bellman, I. Glicksberg, and O. Gross, On the optimal inventory equation, The RAND Corporation, Paper P572.
 50.
 R. Bellman, On some mathematical problems arising in the theory of optimal inventory and stock control, The RAND Corporation, Paper P580.
Additional Information
DOI:
http://dx.doi.org/10.1090/S000299041954098488
PII:
S 00029904(1954)098488
