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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

Vladimir Gaitsgory 1, , Alex Parkinson 1, and  Ilya Shvartsman 2,,

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.

Keywords: Optimal control, discrete systems, infinite horizon, occupational measures, linear programming, duality, numerical methods.
Mathematics Subject Classification: Primary: 49N15, 49M29, 93C55.
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

Figure 1.  The state trajectory - 50 time steps
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Figure 2.  The state trajectory - time step 1
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Figure 3.  The state trajectory - time steps 1 and 2
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Figure 4.  The state trajectory - 50 time steps
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