December  2017, 22(10): 3821-3838. doi: 10.3934/dcdsb.2017192

Linear programming formulations of deterministic infinite horizon optimal control problems 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  January 2017 Revised  March 2017 Published  July 2017

This paper is devoted to a study of infinite horizon optimal control problems with time discounting and time averaging criteria in discrete time. We establish that these problems are related to certain infinite-dimensional linear programming (IDLP) problems. We also establish asymptotic relationships between the optimal values of problems with time discounting and long-run average criteria.

Citation: 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
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 Hamilton-Jacobi-Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1. Google Scholar

[5]

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

[6]

C. J. BishopE. A. Feinberg and J. Zhang, Examples concerning Abel and Cesaro limits, Journal of Mathematical Analysis and Applications, 420 (2014), 1654-1661. doi: 10.1016/j.jmaa.2014.06.017. Google Scholar

[7]

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

[8]

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

[9]

R. BuckdahnD. 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

[10]

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

[11]

N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Wiley & Sons, Inc. , New York, 1988. Google Scholar

[12]

L. FinlayV. 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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

D. Hernandez-HernandezO. Hernandez-Lerma and M. Taksar, The linear programming approach to deterministic optimal control problems, Appl. Math., 24 (1996), 17-33. Google Scholar

[22]

O. Hernandez-Lerma and J. B. Lasserre, The linear programmimg approach, Handbook of Markov Decision Processes: Methods and Applications, 32 (2007), 528-550. doi: 10.1007/978-1-4615-0805-2_12. Google Scholar

[23]

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

[24]

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

[25]

J. B. LasserreD. HenrionC. 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

[26]

E. Lehrer and S. Sorin, A uniform Tauberian theorem in dynamic programming, Mathematics of Operations Research, 17 (1992), 303-307. doi: 10.1287/moor.17.2.303. Google Scholar

[27]

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

[28]

J. Renault, Uniform value in dynamic programming, J. European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254. Google Scholar

[29]

J. E. Rubio, Control and Optimization. The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

R. Vinter, Convex duality and nonlinear optimal control, SIAM J. Control and Optim, 31 (1993), 518-538. doi: 10.1137/0331024. Google Scholar

[34]

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

[35]

A. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006. Google Scholar

[36]

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 Hamilton-Jacobi-Bellman Equations, Systems and Control: Foundations and Applications, Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1. Google Scholar

[5]

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

[6]

C. J. BishopE. A. Feinberg and J. Zhang, Examples concerning Abel and Cesaro limits, Journal of Mathematical Analysis and Applications, 420 (2014), 1654-1661. doi: 10.1016/j.jmaa.2014.06.017. Google Scholar

[7]

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

[8]

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

[9]

R. BuckdahnD. 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

[10]

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

[11]

N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Wiley & Sons, Inc. , New York, 1988. Google Scholar

[12]

L. FinlayV. 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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

D. Hernandez-HernandezO. Hernandez-Lerma and M. Taksar, The linear programming approach to deterministic optimal control problems, Appl. Math., 24 (1996), 17-33. Google Scholar

[22]

O. Hernandez-Lerma and J. B. Lasserre, The linear programmimg approach, Handbook of Markov Decision Processes: Methods and Applications, 32 (2007), 528-550. doi: 10.1007/978-1-4615-0805-2_12. Google Scholar

[23]

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

[24]

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

[25]

J. B. LasserreD. HenrionC. 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

[26]

E. Lehrer and S. Sorin, A uniform Tauberian theorem in dynamic programming, Mathematics of Operations Research, 17 (1992), 303-307. doi: 10.1287/moor.17.2.303. Google Scholar

[27]

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

[28]

J. Renault, Uniform value in dynamic programming, J. European Mathematical Society, 13 (2011), 309-330. doi: 10.4171/JEMS/254. Google Scholar

[29]

J. E. Rubio, Control and Optimization. The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

R. Vinter, Convex duality and nonlinear optimal control, SIAM J. Control and Optim, 31 (1993), 518-538. doi: 10.1137/0331024. Google Scholar

[34]

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

[35]

A. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006. Google Scholar

[36]

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

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