# American Institute of Mathematical Sciences

April  2019, 24(4): 1743-1767. doi: 10.3934/dcdsb.2018235

## Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time

 1 Department of Mathematics, Macquarie University, Macquarie Park, NSW 2113, Australia 2 Department of Mathematics and Computer Science, Penn State Harrisburg, Middletown, PA 17057, USA

* Corresponding author

Received  September 2017 Revised  January 2018 Published  April 2019 Early access  August 2018

It has been recently established that a deterministic infinite horizon discounted optimal control problem in discrete time is closely related to a certain infinite dimensional linear programming problem and its dual, the latter taking the form of a certain max-min problem. In the present paper, we use these results to establish necessary and sufficient optimality conditions for this optimal control problem and to investigate a way how the latter can be used for the construction of a near optimal control.

Citation: Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235
##### References:
 [1] D. Adelman and D. Klabjan, Duality and existence of optimal policies in generalized joint replenishment, Mathematics of Operations Research, 30 (2005), 28-50.  doi: 10.1287/moor.1040.0109.  Google Scholar [2] R. Ash, Measure, Integration and Functional Analysis, Academic Press, 1972.   Google Scholar [3] J.-P. Aubin, Viability Theory, Birkhäuser, 1991.  Google Scholar [4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of HamiltonJacobi-Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar [5] D. Bertsekas, Dynamic Programming and Optimal Control, Athena Scientific, Belmont, MA, 2017.  Google Scholar [6] A. G. Bhatt and V. S. Borkar, Occupation measures for controlled Markov processes: Characterization and optimality, Annals of Probability, 24 (1996), 1531-1562.  doi: 10.1214/aop/1065725192.  Google Scholar [7] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, 1968.   Google Scholar [8] J. Blot, A Pontryagin principle for infinite-horizon problems under constraints, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 19 (2012), 267-275.  Google Scholar [9] V. S. Borkar, A convex analytic approach to Markov decision processes, Probability Theory and Related Fields, 78 (1988), 583-602.  doi: 10.1007/BF00353877.  Google Scholar [10] R. Buckdahn, D. Goreac and M. Quincampoix, Stochastic optimal control and linear programming approach, Appl. Math. Optim., 63 (2011), 257-276.  doi: 10.1007/s00245-010-9120-y.  Google Scholar [11] D. A. Carlson, A. B. Haurier and A. Leizarowicz, Infinite Horizon Optimal Control. Deterministic and Stochastic Processes, Springer, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar [12] N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Wiley & Sons, Inc., New York, 1988.   Google Scholar [13] L. Finlay, V. Gaitsgory and I. Lebedev, Duality in linear programming problems related to deterministic long run average problems of optimal control, SIAM J. Control and Optimization, 47 (2008), 1667-1700.  doi: 10.1137/060676398.  Google Scholar [14] W. H. Fleming and D. Vermes, Convex duality approach to the optimal control of diffusions, SIAM J. Control Optimization, 27 (1989), 1136-1155.  doi: 10.1137/0327060.  Google Scholar [15] V. Gaitsgory, On representation of the limit occupational measures set of control systems with applications to singularly perturbed control systems, SIAM J. Control and Optimization, 43 (2004), 325-340.  doi: 10.1137/S0363012903424186.  Google Scholar [16] V. Gaitsgory, A. Parkinson and I. Shvartsman, Linear programming formulation of a discrete time infinite horizon optimal control problem with time discounting criterion, Proceedings of 55th IEEE Conference on Decision and Control (CDC), 2016, Las Vegas, USA. Google Scholar [17] V. Gaitsgory, A. Parkinson and I. Shvartsman, Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time, Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 3821-3838.  doi: 10.3934/dcdsb.2017192.  Google Scholar [18] V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting, SIAM J. Control and Optim., 48 (2009), 2480-2512.  doi: 10.1137/070696209.  Google Scholar [19] V. Gaitsgory and M. Quincampoix, On sets of occupational measures generated by a deterministic control system on an infinite time horizon, Nonlinear Analysis (Theory, Methods & Applications), 88 (2013), 27-41.  doi: 10.1016/j.na.2013.03.015.  Google Scholar [20] V. Gaitsgory and S. Rossomakhine, Linear programming approach to deterministic long run average problems of optimal control, SIAM J. of Control and Optimization, 44 (2006), 2006-2037.  doi: 10.1137/040616802.  Google Scholar [21] V. Gaitsgory, S. Rossomakhine and N. Thatcher, Approximate solution of the HJB inequality related to the infinite horizon optimal control problem with discounting, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 19 (2012), 65-92.   Google Scholar [22] D. Goreac and O.-S. Serea, Linearization techniques for $L^{∞}$ - control problems and dynamic programming principles in classical and $L^{∞}$ control problems, ESAIM: Control, Optimization and Calculus of Variations, 18 (2012), 836-855.  doi: 10.1051/cocv/2011183.  Google Scholar [23] L. Grüne, Asymptotic controllability and exponential stabilization of nonlinear control systems at singular points, SIAM J. Control Optim., 36 (1998), 1495-1503.  doi: 10.1137/S0363012997315919.  Google Scholar [24] L. Grüne, On the relation between discounted and average optimal value functions, J. Diff. Equations, 148 (1998), 65-69.  doi: 10.1006/jdeq.1998.3451.  Google Scholar [25] D. Hernandez-Hernandez, O. Hernandez-Lerma and M. Taksar, The linear programming approach to deterministic optimal control problems, Appl. Math., 24 (1996), 17-33.  doi: 10.4064/am-24-1-17-33.  Google Scholar [26] O. Hernandez-Lerma and J. B. Lasserre, The linear Programmimg Approach, in the volume Handbook of Markov Decision Processes: Methods and Applications, Edited by E. A. Feinberg and A. Shwartz, Springer, 2012. Google Scholar [27] A. Kamoutsi, T. Sutter, P. M. Esfahani and J. Lygeros, On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems, IEEE Control System Letters, 1 (2017), 134-139.  doi: 10.1109/LCSYS.2017.2710234.  Google Scholar [28] D. Klabjan and D. Adelman, An Infinite-dimensional linear programming algorithm for deterministic semi-Markov decision processes on Borel spaces, Mathematics of Operations Research, 32 (2007), 528-550.  doi: 10.1287/moor.1070.0252.  Google Scholar [29] T. G. Kurtz and R. H. Stockbridge, Existence of Markov controls and characterization of optimal Markov controls, SIAM J. on Control and Optimization, 36 (1998), 609-653.  doi: 10.1137/S0363012995295516.  Google Scholar [30] J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat, Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM J. Control Optim., 47 (2008), 1643-1666.  doi: 10.1137/070685051.  Google Scholar [31] M. Quincampoix and O. Serea, The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures, Nonlinear Anal., 72 (2010), 2803-2815.  doi: 10.1016/j.na.2009.11.024.  Google Scholar [32] J. E. Rubio, Control and Optimization. The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986.   Google Scholar [33] R. H. Stockbridge, Time-Average control of a martingale problem. Existence of a stationary solution, Annals of Probability, 18 (1990), 190-205.  doi: 10.1214/aop/1176990944.  Google Scholar [34] R. H. Stockbridge, Time-average control of a martingale problem: A linear programming formulation, Annals of Probability, 18 (1990), 206-217.  doi: 10.1214/aop/1176990945.  Google Scholar [35] R. Sznajder and J. A. Filar, Some comments on a theorem of Hardy and Littlewood, J. Optimization Theory and Applications, 75 (1992), 201-208.  doi: 10.1007/BF00939913.  Google Scholar [36] R. Vinter, Convex duality and nonlinear optimal control, SIAM J. Control and Optim., 31 (1993), 518-538.  doi: 10.1137/0331024.  Google Scholar [37] A. Zaslavski, Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems, Springer, (2014).  doi: 10.1007/978-3-319-08034-5.  Google Scholar [38] A. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006.   Google Scholar [39] A. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer Optimization and Its Applications, New York, 2014. doi: 10.1007/978-3-319-08828-0.  Google Scholar

show all references

##### References:
 [1] D. Adelman and D. Klabjan, Duality and existence of optimal policies in generalized joint replenishment, Mathematics of Operations Research, 30 (2005), 28-50.  doi: 10.1287/moor.1040.0109.  Google Scholar [2] R. Ash, Measure, Integration and Functional Analysis, Academic Press, 1972.   Google Scholar [3] J.-P. Aubin, Viability Theory, Birkhäuser, 1991.  Google Scholar [4] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of HamiltonJacobi-Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar [5] D. Bertsekas, Dynamic Programming and Optimal Control, Athena Scientific, Belmont, MA, 2017.  Google Scholar [6] A. G. Bhatt and V. S. Borkar, Occupation measures for controlled Markov processes: Characterization and optimality, Annals of Probability, 24 (1996), 1531-1562.  doi: 10.1214/aop/1065725192.  Google Scholar [7] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, 1968.   Google Scholar [8] J. Blot, A Pontryagin principle for infinite-horizon problems under constraints, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 19 (2012), 267-275.  Google Scholar [9] V. S. Borkar, A convex analytic approach to Markov decision processes, Probability Theory and Related Fields, 78 (1988), 583-602.  doi: 10.1007/BF00353877.  Google Scholar [10] R. Buckdahn, D. Goreac and M. Quincampoix, Stochastic optimal control and linear programming approach, Appl. Math. Optim., 63 (2011), 257-276.  doi: 10.1007/s00245-010-9120-y.  Google Scholar [11] D. A. Carlson, A. B. Haurier and A. Leizarowicz, Infinite Horizon Optimal Control. Deterministic and Stochastic Processes, Springer, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar [12] N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Wiley & Sons, Inc., New York, 1988.   Google Scholar [13] L. Finlay, V. Gaitsgory and I. Lebedev, Duality in linear programming problems related to deterministic long run average problems of optimal control, SIAM J. Control and Optimization, 47 (2008), 1667-1700.  doi: 10.1137/060676398.  Google Scholar [14] W. H. Fleming and D. Vermes, Convex duality approach to the optimal control of diffusions, SIAM J. Control Optimization, 27 (1989), 1136-1155.  doi: 10.1137/0327060.  Google Scholar [15] V. Gaitsgory, On representation of the limit occupational measures set of control systems with applications to singularly perturbed control systems, SIAM J. Control and Optimization, 43 (2004), 325-340.  doi: 10.1137/S0363012903424186.  Google Scholar [16] V. Gaitsgory, A. Parkinson and I. Shvartsman, Linear programming formulation of a discrete time infinite horizon optimal control problem with time discounting criterion, Proceedings of 55th IEEE Conference on Decision and Control (CDC), 2016, Las Vegas, USA. Google Scholar [17] V. Gaitsgory, A. Parkinson and I. Shvartsman, Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time, Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 3821-3838.  doi: 10.3934/dcdsb.2017192.  Google Scholar [18] V. Gaitsgory and M. Quincampoix, Linear programming approach to deterministic infinite horizon optimal control problems with discounting, SIAM J. Control and Optim., 48 (2009), 2480-2512.  doi: 10.1137/070696209.  Google Scholar [19] V. Gaitsgory and M. Quincampoix, On sets of occupational measures generated by a deterministic control system on an infinite time horizon, Nonlinear Analysis (Theory, Methods & Applications), 88 (2013), 27-41.  doi: 10.1016/j.na.2013.03.015.  Google Scholar [20] V. Gaitsgory and S. Rossomakhine, Linear programming approach to deterministic long run average problems of optimal control, SIAM J. of Control and Optimization, 44 (2006), 2006-2037.  doi: 10.1137/040616802.  Google Scholar [21] V. Gaitsgory, S. Rossomakhine and N. Thatcher, Approximate solution of the HJB inequality related to the infinite horizon optimal control problem with discounting, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 19 (2012), 65-92.   Google Scholar [22] D. Goreac and O.-S. Serea, Linearization techniques for $L^{∞}$ - control problems and dynamic programming principles in classical and $L^{∞}$ control problems, ESAIM: Control, Optimization and Calculus of Variations, 18 (2012), 836-855.  doi: 10.1051/cocv/2011183.  Google Scholar [23] L. Grüne, Asymptotic controllability and exponential stabilization of nonlinear control systems at singular points, SIAM J. Control Optim., 36 (1998), 1495-1503.  doi: 10.1137/S0363012997315919.  Google Scholar [24] L. Grüne, On the relation between discounted and average optimal value functions, J. Diff. Equations, 148 (1998), 65-69.  doi: 10.1006/jdeq.1998.3451.  Google Scholar [25] D. Hernandez-Hernandez, O. Hernandez-Lerma and M. Taksar, The linear programming approach to deterministic optimal control problems, Appl. Math., 24 (1996), 17-33.  doi: 10.4064/am-24-1-17-33.  Google Scholar [26] O. Hernandez-Lerma and J. B. Lasserre, The linear Programmimg Approach, in the volume Handbook of Markov Decision Processes: Methods and Applications, Edited by E. A. Feinberg and A. Shwartz, Springer, 2012. Google Scholar [27] A. Kamoutsi, T. Sutter, P. M. Esfahani and J. Lygeros, On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems, IEEE Control System Letters, 1 (2017), 134-139.  doi: 10.1109/LCSYS.2017.2710234.  Google Scholar [28] D. Klabjan and D. Adelman, An Infinite-dimensional linear programming algorithm for deterministic semi-Markov decision processes on Borel spaces, Mathematics of Operations Research, 32 (2007), 528-550.  doi: 10.1287/moor.1070.0252.  Google Scholar [29] T. G. Kurtz and R. H. Stockbridge, Existence of Markov controls and characterization of optimal Markov controls, SIAM J. on Control and Optimization, 36 (1998), 609-653.  doi: 10.1137/S0363012995295516.  Google Scholar [30] J. B. Lasserre, D. Henrion, C. Prieur and E. Trélat, Nonlinear optimal control via occupation measures and LMI-relaxations, SIAM J. Control Optim., 47 (2008), 1643-1666.  doi: 10.1137/070685051.  Google Scholar [31] M. Quincampoix and O. Serea, The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures, Nonlinear Anal., 72 (2010), 2803-2815.  doi: 10.1016/j.na.2009.11.024.  Google Scholar [32] J. E. Rubio, Control and Optimization. The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986.   Google Scholar [33] R. H. Stockbridge, Time-Average control of a martingale problem. Existence of a stationary solution, Annals of Probability, 18 (1990), 190-205.  doi: 10.1214/aop/1176990944.  Google Scholar [34] R. H. Stockbridge, Time-average control of a martingale problem: A linear programming formulation, Annals of Probability, 18 (1990), 206-217.  doi: 10.1214/aop/1176990945.  Google Scholar [35] R. Sznajder and J. A. Filar, Some comments on a theorem of Hardy and Littlewood, J. Optimization Theory and Applications, 75 (1992), 201-208.  doi: 10.1007/BF00939913.  Google Scholar [36] R. Vinter, Convex duality and nonlinear optimal control, SIAM J. Control and Optim., 31 (1993), 518-538.  doi: 10.1137/0331024.  Google Scholar [37] A. Zaslavski, Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems, Springer, (2014).  doi: 10.1007/978-3-319-08034-5.  Google Scholar [38] A. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006.   Google Scholar [39] A. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer Optimization and Its Applications, New York, 2014. doi: 10.1007/978-3-319-08828-0.  Google Scholar
The state trajectory - 50 time steps
The state trajectory - time step 1
The state trajectory - time steps 1 and 2
The state trajectory - 50 time steps
 [1] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3821-3838. doi: 10.3934/dcdsb.2017192 [2] Fabio Bagagiolo. An infinite horizon optimal control problem for some switching systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 443-462. doi: 10.3934/dcdsb.2001.1.443 [3] Jingang Zhao, Chi Zhang. Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1471-1483. doi: 10.3934/jimo.2020030 [4] Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 [5] Valery Y. Glizer, Oleg Kelis. Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 211-235. doi: 10.3934/naco.2018013 [6] Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022 [7] Naïla Hayek. Infinite-horizon multiobjective optimal control problems for bounded processes. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1121-1141. doi: 10.3934/dcdss.2018064 [8] Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764 [9] Evelyn Herberg, Michael Hinze, Henrik Schumacher. Maximal discrete sparsity in parabolic optimal control with measures. Mathematical Control & Related Fields, 2020, 10 (4) : 735-759. doi: 10.3934/mcrf.2020018 [10] Senda Ounaies, Jean-Marc Bonnisseau, Souhail Chebbi, Halil Mete Soner. Merton problem in an infinite horizon and a discrete time with frictions. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1323-1331. doi: 10.3934/jimo.2016.12.1323 [11] 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 [12] Alexander Tarasyev, Anastasia Usova. Application of a nonlinear stabilizer for localizing search of optimal trajectories in control problems with infinite horizon. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 389-406. doi: 10.3934/naco.2013.3.389 [13] Yanqun Liu. Duality in linear programming: From trichotomy to quadrichotomy. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1003-1011. doi: 10.3934/jimo.2011.7.1003 [14] Yadong Shu, Bo Li. Linear-quadratic optimal control for discrete-time stochastic descriptor systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021034 [15] Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092 [16] Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009 [17] Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053 [18] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [19] Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial & Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15 [20] Qiying Hu, Wuyi Yue. Optimal control for resource allocation in discrete event systems. Journal of Industrial & Management Optimization, 2006, 2 (1) : 63-80. doi: 10.3934/jimo.2006.2.63

2020 Impact Factor: 1.327