Abstract
This paper deals with infinite horizon, dynamic programs, stated in discrete time, and afflicted by no uncertainty. The essential objective, to be minimized, is the accumulated value of all discounted future costs, and it is assumed to satisfy the crucial condition that every lower level set is bounded with respect to a certain norm. That norm, as well as the natural space of trajectories, is problem intrinsic.
In contrast to standard Markov decision processes (MDP) we admit unbounded singleperiod cost functions and exponential growth within an unlimited state space. Also, no assumption about stationarity in problem data is made.
We show, under broad hypotheses, that any minimizing sequence accumulates to points which solve the dynamic program optimally. This result is important for the study of approximation schemes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.C. Bean and R.L. Smith, Conditions for the existence of planning horizons, Math. Oper. Res. 9 (1984) 391–401.
J.C. Bean and R.L. Smith, Conditions for the discovery of solution horizons, Tech. Rep. 23, Univ. of Michigan, College of Engineering (1986).
M.J. Beckmann, Optimal consumption plans — A dynamic programming approach,New Quantitative Techniques for Economic Analysis (Academic Press, 1982) pp. 183–198.
D.P. Bertsekas,Dynamic Programming (Prentice-Hall, 1987).
E.C. Denardo,Dynamic Programming (Prentice-Hall, 1982).
A.V. Buchvalov and G.J. Lozanovski, On sets closed with respect to convergence in measure in spaces of measurable functions, Sov. Math. Dokl. 14 (1973) 1563–1565.
J. Diestel,Sequences and Series in Banach Spaces, Graduate Texts in Math. (Springer, New York, 1984).
S.D. Flåm and R.J.-B. Wets, Existence results and finite horizon approximates for infinite horizon optimization problems, Econometrica 55 (1987) 1187–1209.
A. Fougères, Optimisation convex dans les Banach non-reflexifs; Méthode de relaxation-projection, Seminaire Choquet (1978–79) (Invitation à l'Analyse).
R.C. Grinold, Convex infinite horizon programs, Math. Progr. 25 (1983) 64–82.
V.L. Levin, Problèmes extrémaux de fonctionelles intégrales semi-continues infériereurement en mesure, Sov. Math. Dokl. 224 (1975) 1256.
R. Mendelsohn and M.J. Sobel, Capital accumulation and the optimization of renewable resource models, J. Econ. Theory 23 (1980) 243–260.
T. Mitra and H.Y. Wan Jr., Some theoretical results on the economics of forestry, Rev. Econ. Studies (1985) 263–282.
R.T. Rockafellar, Integrals which are convex functionals, Pacific J. Math. 39 (1971) 439–469.
R.T. Rockafellar,Convex analysis (Princeton Univ. Press, Princeton, 1970).
R.T. Rockafellar and R.J.-B. Wets, Deterministic and stochastic optimization problems of Bolza type in discrete time, Stochastics 10 (1983).
I.E. Schochetman and R.L. Smith, Infinite horizon optimization, Math. Oper. Res. 14 (1989) 559–574.
I.E. Schochetman and R.L. Smith, Finite dimensional approximation in infinite dimensional mathematical programming, Tech. Rep. 10, Univ. of Michigan, College of Engineering, Ann Arbor (1989).
L. Schwartz,Analyse (Hermann, Paris, 1970).
M. Valadier, Convergence en mesure et optimisation (d'après Levin), Travaux Sém. Analyse Convexe, Univ. de Montpellier, Mathem. no. 14 (1976).
F.A. Valentine,Convex sets (McGraw-Hill, New York, 1964).
M. Weitzman, Duality theory for infinite horizon convex models, Manag. Sci. 19 (1973) 783–789.
A.C. Zaanen,Integration (North-Holland, Amsterdam, 1967).
Author information
Authors and Affiliations
Additional information
Supported by grants from Total Marine via NTNF, and Wilhelm Keilhau's Minnefond.
Rights and permissions
About this article
Cite this article
Flåm, S.D., Fougères, A. Infinite horizon programs; convergence of approximate solutions. Ann Oper Res 29, 333–350 (1991). https://doi.org/10.1007/BF02283604
Issue Date:
DOI: https://doi.org/10.1007/BF02283604