{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T04:28:16Z","timestamp":1773030496604,"version":"3.50.1"},"reference-count":58,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,10,6]],"date-time":"2020-10-06T00:00:00Z","timestamp":1601942400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,10,6]],"date-time":"2020-10-06T00:00:00Z","timestamp":1601942400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005713","name":"Technische Universit\u00e4t M\u00fcnchen","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100005713","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2022,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we propose a semi-metric for Markov processes that allows to bound optimal values of linear Markovian stochastic optimization problems. Similar to existing notions of distance for general stochastic processes, our distance is based on transportation metrics. As opposed to the extant literature, the proposed distance is problem specific, i.e., dependent on the data of the problem whose objective value we want to bound. As a result, we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature. We derive several properties of the proposed semi-metric and demonstrate its use in a stylized numerical example.<\/jats:p>","DOI":"10.1007\/s10107-020-01573-3","type":"journal-article","created":{"date-parts":[[2020,10,6]],"date-time":"2020-10-06T20:02:51Z","timestamp":1602014571000},"page":"871-906","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A stability result for linear Markovian stochastic optimization problems"],"prefix":"10.1007","volume":"191","author":[{"given":"Adriana","family":"Kiszka","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6656-9529","authenticated-orcid":false,"given":"David","family":"Wozabal","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,10,6]]},"reference":[{"issue":"1","key":"1573_CR1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9947-1983-0712247-X","volume":"280","author":"H Attouch","year":"1983","unstructured":"Attouch, H., Wets, R.J.-B.: A convergence theory for saddle functions. Trans. Am. Math. Soc. 280(1), 1\u201341 (1983)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1","key":"1573_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0304-4149(03)00026-7","volume":"106","author":"V Bally","year":"2003","unstructured":"Bally, V., Pag\u00e8s, P.: Error analysis of the quantization algorithm for obstacle problems. Stoch. Process. Their Appl. 106(1), 1\u201340 (2003)","journal-title":"Stoch. Process. Their Appl."},{"issue":"6","key":"1573_CR3","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.3150\/bj\/1072215199","volume":"9","author":"V Bally","year":"2003","unstructured":"Bally, V., Pag\u00e8s, P.: A quantization algorithm for solving discrete time multi-dimensional optimal stopping problems. Bernoulli 9(6), 1003\u20131049 (2003)","journal-title":"Bernoulli"},{"issue":"3","key":"1573_CR4","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1109\/TAC.1975.1100984","volume":"20","author":"D Bertsekas","year":"1981","unstructured":"Bertsekas, D.: Convergence of discretization procedures in dynamic programming. IEEE Trans. Autom. Control 20(3), 415\u2013419 (1981)","journal-title":"IEEE Trans. Autom. Control"},{"key":"1573_CR5","volume-title":"Abstract Dynamic Programming","author":"DP Bertsekas","year":"2013","unstructured":"Bertsekas, D.P.: Abstract Dynamic Programming. Athena Scientific, Belmont (2013)"},{"key":"1573_CR6","volume-title":"Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering","author":"DP Bertsekas","year":"1978","unstructured":"Bertsekas, D.P., Shreve, S.E.: Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering. Academic Press, London (1978)"},{"key":"1573_CR7","volume-title":"Introduction to Stochastic Programming. Springer Series in Operations Research and Financial Engineering","author":"JR Birge","year":"2000","unstructured":"Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer Series in Operations Research and Financial Engineering. Springer, Berlin (2000)"},{"key":"1573_CR8","doi-asserted-by":"crossref","unstructured":"Bourbaki, N., Eggleston, H.G., Madan, S.: Topological Vector Spaces. Springer, \u00c9l\u00e9ments de math\u00e9matique (1987)","DOI":"10.1007\/978-3-642-61715-7"},{"issue":"4","key":"1573_CR9","doi-asserted-by":"crossref","first-page":"35","DOI":"10.21314\/JCF.2004.117","volume":"7","author":"M Broadie","year":"2004","unstructured":"Broadie, M., Glasserman, P.: A stochastic mesh method for pricing high-dimensional american options. J. Comput. Finance 7(4), 35\u201372 (2004)","journal-title":"J. Comput. Finance"},{"key":"1573_CR10","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1287\/moor.1050.0146","volume":"30","author":"MS Casey","year":"2005","unstructured":"Casey, M.S., Sen, S.: The scenario generation algorithm for multistage stochastic linear programming. Math. Oper. Res. 30, 615\u2013631 (2005)","journal-title":"Math. Oper. Res."},{"issue":"8","key":"1573_CR11","doi-asserted-by":"crossref","first-page":"898","DOI":"10.1109\/9.133184","volume":"36","author":"CS Chow","year":"1991","unstructured":"Chow, C.S., Tsitsiklis, J.N.: An optimal one-way multigrid algorithm for discrete-time stochastic control. IEEE Trans. Autom. Control 36(8), 898\u2013914 (1991)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"1","key":"1573_CR12","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1090\/S0002-9947-1983-0684518-7","volume":"276","author":"S Dolecki","year":"1983","unstructured":"Dolecki, S., Salinetti, G., Wets, R.J.B.: Convergence of functions: equi-semicontinuity. Trans. Am. Math. Soc. 276(1), 409\u2013429 (1983)","journal-title":"Trans. Am. Math. Soc."},{"issue":"2","key":"1573_CR13","doi-asserted-by":"crossref","first-page":"1254","DOI":"10.1016\/j.jmaa.2011.11.015","volume":"388","author":"F Dufour","year":"2012","unstructured":"Dufour, F., Prieto-Rumeau, T.: Approximation of Markov decision processes with general state space. J. Math. Anal. Appl. 388(2), 1254\u20131267 (2012)","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"1573_CR14","doi-asserted-by":"crossref","first-page":"1298","DOI":"10.1137\/120867925","volume":"51","author":"F Dufour","year":"2013","unstructured":"Dufour, F., Prieto-Rumeau, T.: Finite linear programming approximations of constrained discounted Markov decision processes. SIAM J. Control Optim. 51(2), 1298\u20131324 (2013)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"1573_CR15","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1080\/17442508.2014.939979","volume":"87","author":"F Dufour","year":"2015","unstructured":"Dufour, F., Prieto-Rumeau, T.: Approximation of average cost Markov decision processes using empirical distributions and concentration inequalities. Stochastics 87(2), 273\u2013307 (2015)","journal-title":"Stochastics"},{"key":"1573_CR16","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1023\/A:1019206915174","volume":"100","author":"J Dupacov\u00e1","year":"2000","unstructured":"Dupacov\u00e1, J., Consigli, G., Wallace, S.W.: Scenarios for multistage stochastic programs. Ann. Oper. Res. 100, 25\u201353 (2000)","journal-title":"Ann. Oper. Res."},{"issue":"3","key":"1573_CR17","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1007\/s10107-002-0331-0","volume":"95","author":"J Dupacov\u00e1","year":"2003","unstructured":"Dupacov\u00e1, J., Gr\u00f6we-Kuska, N., R\u00f6misch, W.: Scenario reduction in stochastic programming an approach using probability metrics. Math. Program. Ser. B 95(3), 493\u2013511 (2003)","journal-title":"Math. Program. Ser. B"},{"key":"1573_CR18","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1023\/A:1018961525394","volume":"85","author":"NCP Edirisinghe","year":"1996","unstructured":"Edirisinghe, N.C.P.: Bound-based approximations in multistage stochastic programming: nonanticipativity aggregation. Ann. Oper. Res. 85, 103\u2013127 (1996)","journal-title":"Ann. Oper. Res."},{"issue":"3","key":"1573_CR19","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1016\/0022-247X(71)90106-5","volume":"34","author":"BL Fox","year":"1971","unstructured":"Fox, B.L.: Finite-state approximations to denumerable-state dynamic programs. J. Math. Anal. Appl. 34(3), 665\u2013670 (1971)","journal-title":"J. Math. Anal. Appl."},{"key":"1573_CR20","first-page":"277","volume":"75","author":"K Frauendorfer","year":"1996","unstructured":"Frauendorfer, K.: Barycentric scenario trees in convex multistage stochastic programming. Math. Program. 75, 277\u2013293 (1996)","journal-title":"Math. Program."},{"issue":"1","key":"1573_CR21","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s10107-015-0958-2","volume":"159","author":"GA Hanasusanto","year":"2016","unstructured":"Hanasusanto, G.A., Kuhn, D., Wiesemann, W.: A comment on \u201ccomputational complexity of stochastic programming problems\u201d. Math. Program. 159(1), 557\u2013569 (2016)","journal-title":"Math. Program."},{"issue":"2\u20133","key":"1573_CR22","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1021805924152","volume":"24","author":"H Heitsch","year":"2003","unstructured":"Heitsch, H., R\u00f6misch, W.: Scenario reduction algorithms in stochastic programming. Comput. Optim. Appl. 24(2\u20133), 187\u2013206 (2003)","journal-title":"Comput. Optim. Appl."},{"key":"1573_CR23","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s10107-007-0197-2","volume":"118","author":"H Heitsch","year":"2009","unstructured":"Heitsch, H., R\u00f6misch, W.: Scenario tree modeling for multistage stochastic programs. Math. Program. 118, 371\u2013406 (2009)","journal-title":"Math. Program."},{"key":"1573_CR24","first-page":"139","volume":"150","author":"H Heitsch","year":"2011","unstructured":"Heitsch, H., R\u00f6misch, W.: Stability and scenario trees for multistage stochastic programs. Int. Ser. Oper. Res. Manag. Sci. 150, 139\u2013164 (2011)","journal-title":"Int. Ser. Oper. Res. Manag. Sci."},{"key":"1573_CR25","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1137\/050632865","volume":"17","author":"H Heitsch","year":"2006","unstructured":"Heitsch, H., R\u00f6misch, W., Strugarek, C.: Stability of multistage stochastic programs. SIAM J. Optim. 17, 511\u2013525 (2006)","journal-title":"SIAM J. Optim."},{"issue":"1","key":"1573_CR26","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s00186-005-0438-1","volume":"62","author":"K Hinderer","year":"2005","unstructured":"Hinderer, K.: Lipschitz continuity of value functions in Markovian decision processes. Math. Methods Oper. Res. 62(1), 3\u201322 (2005)","journal-title":"Math. Methods Oper. Res."},{"issue":"4","key":"1573_CR27","doi-asserted-by":"crossref","first-page":"263","DOI":"10.6028\/jres.049.027","volume":"49","author":"A Hoffman","year":"1952","unstructured":"Hoffman, A.: On approximate solutions of systems of linear inequalities. J. Res. Natl. Bureau Stand. Sect. B Math. Sci. 49(4), 263\u2013265 (1952)","journal-title":"J. Res. Natl. Bureau Stand. Sect. B Math. Sci."},{"issue":"2","key":"1573_CR28","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1023\/A:1021853807313","volume":"24","author":"K H\u00f8yland","year":"2003","unstructured":"H\u00f8yland, K., Kaut, M., Wallace, S.W.: A heuristic for moment-matching scenario generation. Comput. Optim. Appl. 24(2), 169\u2013185 (2003)","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"1573_CR29","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1287\/mnsc.47.2.295.9834","volume":"47","author":"K H\u00f8yland","year":"2001","unstructured":"H\u00f8yland, K., Wallace, S.W.: Generating scenario decision trees for multistage problems. Manag. Sci. 47(2), 295\u2013307 (2001)","journal-title":"Manag. Sci."},{"issue":"37","key":"1573_CR30","first-page":"199","volume":"1","author":"L Kantorovich","year":"1942","unstructured":"Kantorovich, L.: On the translocation of masses. C.R. Acad. Sci. URSS 1(37), 199\u2013201 (1942)","journal-title":"C.R. Acad. Sci. URSS"},{"issue":"2","key":"1573_CR31","first-page":"257","volume":"3","author":"M Kaut","year":"2007","unstructured":"Kaut, M., Wallace, S.W.: Evaluation of scenario generation methods for stochastic programming. Pac. J. Optim. 3(2), 257\u2013271 (2007)","journal-title":"Pac. J. Optim."},{"key":"1573_CR32","volume-title":"Generalized Bounds for Convex Multistage Stochastic Programs","author":"D Kuhn","year":"2005","unstructured":"Kuhn, D.: Generalized Bounds for Convex Multistage Stochastic Programs. Springer Berlin Heidelberg, Berlin (2005)"},{"issue":"4","key":"1573_CR33","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1287\/moor.6.4.493","volume":"6","author":"H-J Langen","year":"1981","unstructured":"Langen, H.-J.: Convergence of dynamic programming models. Math. Oper. Res. 6(4), 493\u2013512 (1981)","journal-title":"Math. Oper. Res."},{"issue":"4","key":"1573_CR34","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1287\/opre.2013.1182","volume":"61","author":"N L\u00f6hndorf","year":"2013","unstructured":"L\u00f6hndorf, N., Wozabal, D., Minner, S.: Optimizing trading decisions for hydro storage systems using approximate dual dynamic programming. Oper. Res. 61(4), 810\u2013823 (2013)","journal-title":"Oper. Res."},{"key":"1573_CR35","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198514855.001.0001","volume-title":"Introduction to Functional Analysis","author":"R Meise","year":"1997","unstructured":"Meise, R., Vogt, D., Ramanujan, M.S.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)"},{"issue":"4","key":"1573_CR36","doi-asserted-by":"crossref","first-page":"872","DOI":"10.1287\/moor.22.4.872","volume":"22","author":"A M\u00fcller","year":"1997","unstructured":"M\u00fcller, A.: How does the value function of a Markov decision process depend on the transition probabilities? Math. Oper. Res. 22(4), 872\u2013885 (1997)","journal-title":"Math. Oper. Res."},{"issue":"1","key":"1573_CR37","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1287\/moor.1040.0114","volume":"30","author":"T Pennanen","year":"2005","unstructured":"Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs. Math. Oper. Res. 30(1), 245\u2013256 (2005)","journal-title":"Math. Oper. Res."},{"issue":"1\u20132","key":"1573_CR38","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/s10107-007-0113-9","volume":"116","author":"T Pennanen","year":"2009","unstructured":"Pennanen, T.: Epi-convergent discretizations of multistage stochastic programs via integration quadratures. Math. Program. 116(1\u20132), 461\u2013479 (2009)","journal-title":"Math. Program."},{"issue":"2","key":"1573_CR39","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/BF01582895","volume":"52","author":"MVF Pereira","year":"1991","unstructured":"Pereira, M.V.F., Pinto, L.M.V.G.: Multi-stage stochastic optimization applied to energy planning. Math. Program. 52(2), 359\u2013375 (1991)","journal-title":"Math. Program."},{"issue":"2","key":"1573_CR40","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/PL00011398","volume":"89","author":"GC Pflug","year":"2001","unstructured":"Pflug, G.C.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program. Ser. B 89(2), 251\u2013271 (2001)","journal-title":"Math. Program. Ser. B"},{"issue":"1","key":"1573_CR41","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/110825054","volume":"22","author":"GC Pflug","year":"2012","unstructured":"Pflug, G.C., Pichler, A.: A distance for multistage stochastic optimization models. SIAM J. Optim. 22(1), 1\u201323 (2012)","journal-title":"SIAM J. Optim."},{"key":"1573_CR42","volume-title":"Multistage Stochastic Optimization. Springer Series in Operations Research and Financial Engineering","author":"GC Pflug","year":"2014","unstructured":"Pflug, G.C., Pichler, A.: Multistage Stochastic Optimization. Springer Series in Operations Research and Financial Engineering. Springer, Berlin (2014)"},{"issue":"3","key":"1573_CR43","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1137\/15M1043376","volume":"26","author":"GC Pflug","year":"2016","unstructured":"Pflug, G.C., Pichler, A.: From empirical observations to tree models for stochastic optimization: convergence properties. SIAM J. Optim. 26(3), 1715\u20131740 (2016)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"1573_CR44","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1016\/j.orl.2008.01.013","volume":"36","author":"AB Philpott","year":"2008","unstructured":"Philpott, A.B., Guan, Z.: On the convergence of stochastic dual dynamic programming and related methods. Oper. Res. Lett. 36(4), 450\u2013455 (2008)","journal-title":"Oper. Res. Lett."},{"key":"1573_CR45","doi-asserted-by":"crossref","unstructured":"Pollard, D.: A User\u2019s Guide to Measure Theoretic Probability. Cambridge Series in Statistical and Probabilistic Mathematics (2002)","DOI":"10.1017\/CBO9780511811555"},{"key":"1573_CR46","unstructured":"Ren, Z., Krogh, B.H.: State aggregation in Markov decision processes. In: Proceedings of the 41st IEEE Conference on Decision and Control, 2002, vol.\u00a04, pp. 3819\u20133824 (2002)"},{"key":"1573_CR47","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1137\/0712056","volume":"12","author":"SM Robinson","year":"1975","unstructured":"Robinson, S.M.: Stability theory for systems of inequalities. SIAM J. Numer. Anal. 12, 754\u2013769 (1975)","journal-title":"SIAM J. Numer. Anal."},{"key":"1573_CR48","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF02204819","volume":"30","author":"W R\u00f6misch","year":"1991","unstructured":"R\u00f6misch, W., Schultz, R.: Stability analysis for stochastic programs. Ann. Oper. Res. 30, 241\u2013266 (1991)","journal-title":"Ann. Oper. Res."},{"issue":"4","key":"1573_CR49","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1287\/moor.2016.0832","volume":"42","author":"N Saldi","year":"2017","unstructured":"Saldi, N., Y\u00fcksel, S., Linder, T.: On the asymptotic optimality of finite approximations to Markov decision processes with borel spaces. Math. Oper. Res. 42(4), 945\u2013978 (2017)","journal-title":"Math. Oper. Res."},{"issue":"1","key":"1573_CR50","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.orl.2005.02.003","volume":"34","author":"A Shapiro","year":"2006","unstructured":"Shapiro, A.: On complexity of multistage stochastic programs. Oper. Res. Lett. 34(1), 1\u20138 (2006)","journal-title":"Oper. Res. Lett."},{"key":"1573_CR51","unstructured":"Shapiro, A.: Computational complexity of stochastic programming: Monte Carlo sampling approach. In: Proceedings of the International Congress of Mathematicians, pp. 2979\u20132995 (2010)"},{"issue":"1","key":"1573_CR52","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.ejor.2010.08.007","volume":"209","author":"A Shapiro","year":"2011","unstructured":"Shapiro, A.: Analysis of stochastic dual dynamic programming method. Eur. J. Oper. Res. 209(1), 63\u201372 (2011)","journal-title":"Eur. J. Oper. Res."},{"key":"1573_CR53","doi-asserted-by":"crossref","unstructured":"Shapiro, A., Dentcheva, D., Ruszczy\u0144ski, A.: Lectures on Stochastic Programming. MPS-SIAM Series on Optimization 9 (2009)","DOI":"10.1137\/1.9780898718751"},{"key":"1573_CR54","doi-asserted-by":"crossref","unstructured":"Villani, C.: Topics in Optimal Transportation. American Mathematical Society. Graduate Studies in Mathematics, vol. 58 (2003)","DOI":"10.1090\/gsm\/058"},{"key":"1573_CR55","doi-asserted-by":"crossref","unstructured":"Villani, C.: Optimal Transport, Old and New. Springer, vol. 338. Grundlehren der Mathematischen Wissenschaften (2009)","DOI":"10.1007\/978-3-540-71050-9"},{"key":"1573_CR56","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1090\/S0002-9939-1969-0246200-8","volume":"23","author":"D Walkup","year":"1969","unstructured":"Walkup, D., Wets, R.J.B.: A lipschitzian characterization of convex polyhedra. Proc. Am. Math. Soc. 23, 167\u2013173 (1969)","journal-title":"Proc. Am. Math. Soc."},{"key":"1573_CR57","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/0022-247X(80)90128-6","volume":"74","author":"DJ White","year":"1980","unstructured":"White, D.J.: Finite state approximations for denumerable state infinite horizon discounted Markov decision processes. J. Math. Anal. Appl. 74, 292\u2013295 (1980)","journal-title":"J. Math. Anal. Appl."},{"issue":"1","key":"1573_CR58","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/0022-247X(82)90271-2","volume":"86","author":"DJ White","year":"1982","unstructured":"White, D.J.: Finite state approximations for denumerable state infinite horizon discounted Markov decision processes with unbounded rewards. J. Math. Anal. Appl. 86(1), 292\u2013306 (1982)","journal-title":"J. Math. Anal. Appl."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01573-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-020-01573-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01573-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,15]],"date-time":"2024-08-15T15:51:47Z","timestamp":1723737107000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-020-01573-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,6]]},"references-count":58,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["1573"],"URL":"https:\/\/doi.org\/10.1007\/s10107-020-01573-3","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,6]]},"assertion":[{"value":"2 July 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 September 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 October 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}