# American Institute of Mathematical Sciences

• Previous Article
Performance analysis of transmission rate control algorithm from readers to a middleware in intelligent transportation systems
• NACO Home
• This Issue
• Next Article
A sufficient optimality condition for nonregular problems via a nonlinear Lagrangian
2012, 2(2): 333-355. doi: 10.3934/naco.2012.2.333

## Duration problem with multiple exchanges

 1 The University of Adelaide, Department of Applied Mathathematics, Adelaide, South Australia 5005, Australia 2 Institute of Mathematics and Computer Science, Wrocław University of Techechnology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland 3 Department of Business Administration, Aichi University, Nagoya Campus, Hiraike 4-60-6, Nakamura, Nagoya, Aichi 453-8777, Japan

Received  October 2011 Revised  May 2012 Published  May 2012

We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the $m$--choice duration problem, there exists a sequence $(s_1,s_2,\ldots,s_m)$ of critical numbers such that, whenever there remain $k$ choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost.
Citation: Charles E. M. Pearce, Krzysztof Szajowski, Mitsushi Tamaki. Duration problem with multiple exchanges. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 333-355. doi: 10.3934/naco.2012.2.333
##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,", Dover, (2000). Google Scholar [2] K. Ano, Optimal selection problem with three stops,, J. Oper. Res. Soc., 32 (1989), 491. Google Scholar [3] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Ratio prophet inequalities when the mortal has several choices,, Ann. Appl. Probab., 12 (2002), 972. doi: 10.1214/aoap/1031863177. Google Scholar [4] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Two-choice optimal stopping,, Adv. Appl. Probab., 36 (2004), 1116. doi: 10.1239/aap/1103662960. Google Scholar [5] D. Assaf and E. Samuel-Cahn, Simple ratio prophet inequalities for a mortal with multiple choices,, J. Appl. Probab., 37 (2000), 1084. doi: 10.1239/jap/1014843085. Google Scholar [6] Y. Chow, H. Robbins and D. Siegmund, "Great Expectations: The Theory of Optimal Stopping,", Houghton Mifflin Co., (1971). Google Scholar [7] E. Dynkin and A. Yushkevich, "Markov Process, Theorems and Problems,", Plenum Press, (1969). Google Scholar [8] R. Eidukjavicjus, Optimalna ostanovka markovskoj cepi dvumia momentami ostanovki,, Lit. Mat. Sb., 13 (1979), 181. Google Scholar [9] T. Ferguson, Who solved the secretary problem?, Statist. Sci. 4 (1989), 4 (1989), 282. doi: 10.1214/ss/1177012493. Google Scholar [10] T. Ferguson, J. Hardwick and M. Tamaki, Maximizing the duration of owning a relatively best object, Strategies for sequential search and selection in real time (Amherst, MA, 1990)., Contemp. Math., 125 (1992), 37. doi: 10.1090/conm/125/1160608. Google Scholar [11] A. Frank and S. Samuels, On an optimal problem of Gusein-Zade,, Stoch. Proc. Appl., 10 (1980), 299. doi: 10.1016/0304-4149(80)90013-7. Google Scholar [12] J. P. Gilbert and F. Mosteller, Recognizing the maximum of a sequence,, J. Am. Stat. Assoc., 61 (1966), 35. Google Scholar [13] A. V. Gnedin, Best choice from the planar poisson process,, Stoch. Proc. Appl., 111 (2004), 317. doi: 10.1016/j.spa.2003.12.005. Google Scholar [14] A. V. Gnedin, Objectives in the best-choice problems,, Sequential Analysis, 24 (2005), 1. doi: 10.1081/SQA-200056196. Google Scholar [15] I. Gradshteyn and I. Ryzhik, "Tables of Integrals, Series, and Products,", 6th Edition, (2000). Google Scholar [16] S. M. Gusein-Zade, The problem of choice and the optimal stopping rule for a sequence of independent trials,, Theory Probab. Appl., 11 (1966), 472. doi: 10.1137/1111050. Google Scholar [17] G. W. Haggstrom, Optimal sequential procedures when more then one stop is required,, Ann. Math. Stat., 38 (1967), 1618. doi: 10.1214/aoms/1177698595. Google Scholar [18] M. Hu, M. Tamaki and K. Ohno, A continuous time duration problem with an unknown number of options,, Math. Japonica, 48 (1998), 233. Google Scholar [19] A. Kurushima and K. Ano, On optimal stopping problems for possession-period maximization associated with Poisson arrival,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 11. Google Scholar [20] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, J. Appl. Probab., 46 (2009), 402. doi: 10.1239/jap/1245676096. Google Scholar [21] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, 13-th International Symposium on Dyn. Games and Appl., (2008). Google Scholar [22] R. Kühne and L. Rüschendorf, On optimal two-stopping problems,, In, II (2002), 261. Google Scholar [23] A. Lehtinen, The best-choice problem with an unknown number of objects,, Z. Oper.Res., 37 (1993), 97. Google Scholar [24] V. V. Mazalov and M. Tamaki, Explicit solutions to the duration problem,, Aichi Keiei Ronsyu, 147 (2003), 69. Google Scholar [25] V. V. Mazalov and M. Tamaki, An explicit formula for the limiting optimal gain in the full information duration problem,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 217. Google Scholar [26] V. V. Mazalov and M. Tamaki, An explicit formula for the optimal gain in the full-information problem of owning a relatively best object,, J. Appl. Probab., 43 (2006), 87. doi: 10.1239/jap/1143936245. Google Scholar [27] V. V. Mazalov and M. Tamaki, Duration problem on trajectories,, Stochastics, 79 (2007), 211. doi: 10.1080/17442500601072704. Google Scholar [28] T. F. Móri, The random secretary problem with multiple choice,, Ann. Univ. Sci. Budapest. de Rolando Eötvös Nominatae. Sect. Comput., 5 (1984), 91. Google Scholar [29] A. G. Mucci, Differential equations and optimal choice problem,, Ann. Stat., 1 (1973), 104. doi: 10.1214/aos/1193342386. Google Scholar [30] A. G. Mucci, On a class of secretary problems,, Ann. Probab., 1 (1973), 417. doi: 10.1214/aop/1176996936. Google Scholar [31] M. L. Nikolaev, Zadacha vybora dvokh ob"ektov z minimal'nom sumarnym rangom,, Izv. Vyss. Uchebn. Zaved. Mat., 166 (1976), 33. Google Scholar [32] M. L. Nikolaev, Ob odnom obobshchenii zadachi nailuchshego vybora,, Teor. Veroyatn. Primen., 22 (1977), 191. doi: 10.1137/1122023. Google Scholar [33] M. L. Nikolaev, Obobshchennyje posledovatelnyje procedury,, Litov. Mat. Sb., 191 (1979), 35. Google Scholar [34] M. Nikolaev, Optimal multi-stopping rules,, Obozr. Prikl. Prom. Mat., 5 (1998), 309. Google Scholar [35] M. L. Nikolaev, On optimal multiple stopping of Markov sequences,, Teor. Veroyatnost. i Primenen., 43 (1998), 374. Google Scholar [36] J. Petruccelli, On the best-choice problem when the number of observation is random,, J. Appl. Probab., 20 (1983), 165. doi: 10.2307/3213731. Google Scholar [37] Z. Porosiński, The full-information best choice problem with a random number of observations,, Stoch. Proc. Appl., 24 (1987), 293. doi: 10.1016/0304-4149(87)90020-2. Google Scholar [38] Z. Porosinski, On best choice problems having similar solutions,, Stat. Probab. Lett., 56 (2002), 321. doi: 10.1016/S0167-7152(01)00201-2. Google Scholar [39] J. Preater, On multiple choice secretary problem,, Math. Oper. Res., 19 (): 597. doi: 10.1287/moor.19.3.597. Google Scholar [40] E. Presman and I. Sonin, The best choice problem for a random number of objects,, Theory Prob. Appl., 17 (1972), 657. doi: 10.1137/1117078. Google Scholar [41] S. M. Ross, "Applied Probability Models with Optimization Applications,", Holden-Day, (1970). Google Scholar [42] M. P. Quine and J. S. Law, Exact results for a secretary problem,, J. Appl. Probab., 33 (1996), 630. doi: 10.2307/3215345. Google Scholar [43] M. Sakaguchi, Dowry problem and OLA policies,, Rep. Stat. Appl. Res. Union Jap. Sci. Eng. JUSE, 25 (1978), 124. Google Scholar [44] M. Sakaguchi, Generalized secretary problems with three stops,, Math. Japonica, 32 (1987), 105. Google Scholar [45] S. M. Samuels, Secretary problems,, In, (1991), 381. Google Scholar [46] S. M. Samuels, Why do these quite different best-choice problems have the same solutions?, Adv. Appl. Probab., 36 (2004), 398. doi: 10.1239/aap/1086957578. Google Scholar [47] W. Stadje, On multiple stopping rules,, Optimization, 16 (1985), 401. doi: 10.1080/02331938508843030. Google Scholar [48] A. Suchwałko and K. Szajowski, Non standard, no information secretary problems,, Sci. Math. Japonicae, 56 (2002), 443. Google Scholar [49] K. Szajowski, Optimal choice problem of $a$-th object,, Matem. Stos., 19 (1982), 51. Google Scholar [50] K. Szajowski, A two-disorder detection problem,, Applicationes Mathematicae, 24 (1996), 231. Google Scholar [51] K. Szajowski and M. Tamaki, Shelf life of candidates in the generalized secretary problem,, preprint, (2006). Google Scholar [52] K. Szajowski, On a random number of disorders,, Probab. Math. Statist., 31 (2011), 17. Google Scholar [53] K. Szajowski, On stopping games when more than one stop is possible,, In, (2000), 57. Google Scholar [54] M. Tamaki, A secretary problem with double choice,, J. Oper. Res. Soc. Jap., 22 (1979), 257. Google Scholar [55] M. Tamaki, OLA policy and the best choice problem with random number of objects,, Math. Japonica, 24 (1979), 451. Google Scholar [56] M. Tamaki, C. E. Pearce and K. Szajowski, Multiple choice problems related to the duration of the secretary problem,, Sūrikaisekikenkyūsho Kōkyūroku 1068 (1998), 1068 (1998), 75. Google Scholar [57] J. Wilson, Optimal choice and assignment of the best $m$ of $n$ randomly arriving items,, Stoch. Proc. Appl., 39 (1991), 325. doi: 10.1016/0304-4149(91)90086-R. Google Scholar [58] M. Yasuda, On a stopping problem involving refusal and forced stopping,, J. Appl. Probab., 20 (1983), 71. doi: 10.2307/3213721. Google Scholar [59] M. Yasuda and K. Szajowski, Dynkin games and its extension to a multiple stopping model,, Bulletin of the Japan Society for Industrial Mathematics 12 (2002), 12 (2002), 17. Google Scholar [60] A. J. Yeo and G. F. Yeo, Selecting satisfactory secretaries,, Austral. J. Statist., 36 (1994), 185. Google Scholar [61] G. F. Yeo, Duration of a secretary problem,, J. Appl. Probab., 34 (1997), 556. Google Scholar

show all references

##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,", Dover, (2000). Google Scholar [2] K. Ano, Optimal selection problem with three stops,, J. Oper. Res. Soc., 32 (1989), 491. Google Scholar [3] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Ratio prophet inequalities when the mortal has several choices,, Ann. Appl. Probab., 12 (2002), 972. doi: 10.1214/aoap/1031863177. Google Scholar [4] D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Two-choice optimal stopping,, Adv. Appl. Probab., 36 (2004), 1116. doi: 10.1239/aap/1103662960. Google Scholar [5] D. Assaf and E. Samuel-Cahn, Simple ratio prophet inequalities for a mortal with multiple choices,, J. Appl. Probab., 37 (2000), 1084. doi: 10.1239/jap/1014843085. Google Scholar [6] Y. Chow, H. Robbins and D. Siegmund, "Great Expectations: The Theory of Optimal Stopping,", Houghton Mifflin Co., (1971). Google Scholar [7] E. Dynkin and A. Yushkevich, "Markov Process, Theorems and Problems,", Plenum Press, (1969). Google Scholar [8] R. Eidukjavicjus, Optimalna ostanovka markovskoj cepi dvumia momentami ostanovki,, Lit. Mat. Sb., 13 (1979), 181. Google Scholar [9] T. Ferguson, Who solved the secretary problem?, Statist. Sci. 4 (1989), 4 (1989), 282. doi: 10.1214/ss/1177012493. Google Scholar [10] T. Ferguson, J. Hardwick and M. Tamaki, Maximizing the duration of owning a relatively best object, Strategies for sequential search and selection in real time (Amherst, MA, 1990)., Contemp. Math., 125 (1992), 37. doi: 10.1090/conm/125/1160608. Google Scholar [11] A. Frank and S. Samuels, On an optimal problem of Gusein-Zade,, Stoch. Proc. Appl., 10 (1980), 299. doi: 10.1016/0304-4149(80)90013-7. Google Scholar [12] J. P. Gilbert and F. Mosteller, Recognizing the maximum of a sequence,, J. Am. Stat. Assoc., 61 (1966), 35. Google Scholar [13] A. V. Gnedin, Best choice from the planar poisson process,, Stoch. Proc. Appl., 111 (2004), 317. doi: 10.1016/j.spa.2003.12.005. Google Scholar [14] A. V. Gnedin, Objectives in the best-choice problems,, Sequential Analysis, 24 (2005), 1. doi: 10.1081/SQA-200056196. Google Scholar [15] I. Gradshteyn and I. Ryzhik, "Tables of Integrals, Series, and Products,", 6th Edition, (2000). Google Scholar [16] S. M. Gusein-Zade, The problem of choice and the optimal stopping rule for a sequence of independent trials,, Theory Probab. Appl., 11 (1966), 472. doi: 10.1137/1111050. Google Scholar [17] G. W. Haggstrom, Optimal sequential procedures when more then one stop is required,, Ann. Math. Stat., 38 (1967), 1618. doi: 10.1214/aoms/1177698595. Google Scholar [18] M. Hu, M. Tamaki and K. Ohno, A continuous time duration problem with an unknown number of options,, Math. Japonica, 48 (1998), 233. Google Scholar [19] A. Kurushima and K. Ano, On optimal stopping problems for possession-period maximization associated with Poisson arrival,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 11. Google Scholar [20] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, J. Appl. Probab., 46 (2009), 402. doi: 10.1239/jap/1245676096. Google Scholar [21] A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, 13-th International Symposium on Dyn. Games and Appl., (2008). Google Scholar [22] R. Kühne and L. Rüschendorf, On optimal two-stopping problems,, In, II (2002), 261. Google Scholar [23] A. Lehtinen, The best-choice problem with an unknown number of objects,, Z. Oper.Res., 37 (1993), 97. Google Scholar [24] V. V. Mazalov and M. Tamaki, Explicit solutions to the duration problem,, Aichi Keiei Ronsyu, 147 (2003), 69. Google Scholar [25] V. V. Mazalov and M. Tamaki, An explicit formula for the limiting optimal gain in the full information duration problem,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 217. Google Scholar [26] V. V. Mazalov and M. Tamaki, An explicit formula for the optimal gain in the full-information problem of owning a relatively best object,, J. Appl. Probab., 43 (2006), 87. doi: 10.1239/jap/1143936245. Google Scholar [27] V. V. Mazalov and M. Tamaki, Duration problem on trajectories,, Stochastics, 79 (2007), 211. doi: 10.1080/17442500601072704. Google Scholar [28] T. F. Móri, The random secretary problem with multiple choice,, Ann. Univ. Sci. Budapest. de Rolando Eötvös Nominatae. Sect. Comput., 5 (1984), 91. Google Scholar [29] A. G. Mucci, Differential equations and optimal choice problem,, Ann. Stat., 1 (1973), 104. doi: 10.1214/aos/1193342386. Google Scholar [30] A. G. Mucci, On a class of secretary problems,, Ann. Probab., 1 (1973), 417. doi: 10.1214/aop/1176996936. Google Scholar [31] M. L. Nikolaev, Zadacha vybora dvokh ob"ektov z minimal'nom sumarnym rangom,, Izv. Vyss. Uchebn. Zaved. Mat., 166 (1976), 33. Google Scholar [32] M. L. Nikolaev, Ob odnom obobshchenii zadachi nailuchshego vybora,, Teor. Veroyatn. Primen., 22 (1977), 191. doi: 10.1137/1122023. Google Scholar [33] M. L. Nikolaev, Obobshchennyje posledovatelnyje procedury,, Litov. Mat. Sb., 191 (1979), 35. Google Scholar [34] M. Nikolaev, Optimal multi-stopping rules,, Obozr. Prikl. Prom. Mat., 5 (1998), 309. Google Scholar [35] M. L. Nikolaev, On optimal multiple stopping of Markov sequences,, Teor. Veroyatnost. i Primenen., 43 (1998), 374. Google Scholar [36] J. Petruccelli, On the best-choice problem when the number of observation is random,, J. Appl. Probab., 20 (1983), 165. doi: 10.2307/3213731. Google Scholar [37] Z. Porosiński, The full-information best choice problem with a random number of observations,, Stoch. Proc. Appl., 24 (1987), 293. doi: 10.1016/0304-4149(87)90020-2. Google Scholar [38] Z. Porosinski, On best choice problems having similar solutions,, Stat. Probab. Lett., 56 (2002), 321. doi: 10.1016/S0167-7152(01)00201-2. Google Scholar [39] J. Preater, On multiple choice secretary problem,, Math. Oper. Res., 19 (): 597. doi: 10.1287/moor.19.3.597. Google Scholar [40] E. Presman and I. Sonin, The best choice problem for a random number of objects,, Theory Prob. Appl., 17 (1972), 657. doi: 10.1137/1117078. Google Scholar [41] S. M. Ross, "Applied Probability Models with Optimization Applications,", Holden-Day, (1970). Google Scholar [42] M. P. Quine and J. S. Law, Exact results for a secretary problem,, J. Appl. Probab., 33 (1996), 630. doi: 10.2307/3215345. Google Scholar [43] M. Sakaguchi, Dowry problem and OLA policies,, Rep. Stat. Appl. Res. Union Jap. Sci. Eng. JUSE, 25 (1978), 124. Google Scholar [44] M. Sakaguchi, Generalized secretary problems with three stops,, Math. Japonica, 32 (1987), 105. Google Scholar [45] S. M. Samuels, Secretary problems,, In, (1991), 381. Google Scholar [46] S. M. Samuels, Why do these quite different best-choice problems have the same solutions?, Adv. Appl. Probab., 36 (2004), 398. doi: 10.1239/aap/1086957578. Google Scholar [47] W. Stadje, On multiple stopping rules,, Optimization, 16 (1985), 401. doi: 10.1080/02331938508843030. Google Scholar [48] A. Suchwałko and K. Szajowski, Non standard, no information secretary problems,, Sci. Math. Japonicae, 56 (2002), 443. Google Scholar [49] K. Szajowski, Optimal choice problem of $a$-th object,, Matem. Stos., 19 (1982), 51. Google Scholar [50] K. Szajowski, A two-disorder detection problem,, Applicationes Mathematicae, 24 (1996), 231. Google Scholar [51] K. Szajowski and M. Tamaki, Shelf life of candidates in the generalized secretary problem,, preprint, (2006). Google Scholar [52] K. Szajowski, On a random number of disorders,, Probab. Math. Statist., 31 (2011), 17. Google Scholar [53] K. Szajowski, On stopping games when more than one stop is possible,, In, (2000), 57. Google Scholar [54] M. Tamaki, A secretary problem with double choice,, J. Oper. Res. Soc. Jap., 22 (1979), 257. Google Scholar [55] M. Tamaki, OLA policy and the best choice problem with random number of objects,, Math. Japonica, 24 (1979), 451. Google Scholar [56] M. Tamaki, C. E. Pearce and K. Szajowski, Multiple choice problems related to the duration of the secretary problem,, Sūrikaisekikenkyūsho Kōkyūroku 1068 (1998), 1068 (1998), 75. Google Scholar [57] J. Wilson, Optimal choice and assignment of the best $m$ of $n$ randomly arriving items,, Stoch. Proc. Appl., 39 (1991), 325. doi: 10.1016/0304-4149(91)90086-R. Google Scholar [58] M. Yasuda, On a stopping problem involving refusal and forced stopping,, J. Appl. Probab., 20 (1983), 71. doi: 10.2307/3213721. Google Scholar [59] M. Yasuda and K. Szajowski, Dynkin games and its extension to a multiple stopping model,, Bulletin of the Japan Society for Industrial Mathematics 12 (2002), 12 (2002), 17. Google Scholar [60] A. J. Yeo and G. F. Yeo, Selecting satisfactory secretaries,, Austral. J. Statist., 36 (1994), 185. Google Scholar [61] G. F. Yeo, Duration of a secretary problem,, J. Appl. Probab., 34 (1997), 556. Google Scholar
 [1] Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108 [2] Dong Li, Tong Li. Shock formation in a traffic flow model with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2011, 6 (4) : 681-694. doi: 10.3934/nhm.2011.6.681 [3] Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1 [4] Wenqing Bao, Xianyi Wu, Xian Zhou. Optimal stopping problems with restricted stopping times. Journal of Industrial & Management Optimization, 2017, 13 (1) : 399-411. doi: 10.3934/jimo.2016023 [5] Ryan Loxton, Qun Lin. Optimal fleet composition via dynamic programming and golden section search. Journal of Industrial & Management Optimization, 2011, 7 (4) : 875-890. doi: 10.3934/jimo.2011.7.875 [6] Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431 [7] Yongki Lee, Hailiang Liu. Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 323-339. doi: 10.3934/dcds.2015.35.323 [8] Kevin Kuo, Phong Luu, Duy Nguyen, Eric Perkerson, Katherine Thompson, Qing Zhang. Pairs trading: An optimal selling rule. Mathematical Control & Related Fields, 2015, 5 (3) : 489-499. doi: 10.3934/mcrf.2015.5.489 [9] Haiying Liu, Wenjie Bi, Kok Lay Teo, Naxing Liu. Dynamic optimal decision making for manufacturers with limited attention based on sparse dynamic programming. Journal of Industrial & Management Optimization, 2019, 15 (2) : 445-464. doi: 10.3934/jimo.2018050 [10] Jakob Kotas. Optimal stopping for response-guided dosing. Networks & Heterogeneous Media, 2019, 14 (1) : 43-52. doi: 10.3934/nhm.2019003 [11] Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261 [12] Yan-An Hwang, Yu-Hsien Liao. Reduction and dynamic approach for the multi-choice Shapley value. Journal of Industrial & Management Optimization, 2013, 9 (4) : 885-892. doi: 10.3934/jimo.2013.9.885 [13] Hoi Tin Kong, Qing Zhang. An optimal trading rule of a mean-reverting asset. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1403-1417. doi: 10.3934/dcdsb.2010.14.1403 [14] Jingzhi Tie, Qing Zhang. Switching between a pair of stocks: An optimal trading rule. Mathematical Control & Related Fields, 2018, 8 (3&4) : 965-999. doi: 10.3934/mcrf.2018042 [15] Yoshikazu Giga, Hirotoshi Kuroda. A counterexample to finite time stopping property for one-harmonic map flow. Communications on Pure & Applied Analysis, 2015, 14 (1) : 121-125. doi: 10.3934/cpaa.2015.14.121 [16] Haibo Jin, Long Hai, Xiaoliang Tang. An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2018188 [17] Zihui Liu, Xiangyong Zeng. The geometric structure of relative one-weight codes. Advances in Mathematics of Communications, 2016, 10 (2) : 367-377. doi: 10.3934/amc.2016011 [18] Jin Ma, Xinyang Wang, Jianfeng Zhang. Dynamic equilibrium limit order book model and optimal execution problem. Mathematical Control & Related Fields, 2015, 5 (3) : 557-583. doi: 10.3934/mcrf.2015.5.557 [19] Mou-Hsiung Chang, Tao Pang, Moustapha Pemy. Finite difference approximation for stochastic optimal stopping problems with delays. Journal of Industrial & Management Optimization, 2008, 4 (2) : 227-246. doi: 10.3934/jimo.2008.4.227 [20] Andrzej Nowakowski, Jan Sokolowski. On dual dynamic programming in shape control. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2473-2485. doi: 10.3934/cpaa.2012.11.2473

Impact Factor: