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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 |
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.
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. |
| [2] |
R. Ash, Measure, Integration and Functional Analysis, Academic Press, 1972. |
| [3] |
J. -P. Aubin,
Viability Theory Birkhäuser, 1991. |
| [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. |
| [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. |
| [6] |
C. J. Bishop, E. 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. |
| [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.
|
| [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. |
| [9] |
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. |
| [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. |
| [11] |
N. Dunford and J. T. Schwartz,
Linear Operators, Part I, General Theory, Wiley & Sons, Inc. , New York, 1988. |
| [12] |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [21] |
D. Hernandez-Hernandez, O. Hernandez-Lerma and M. Taksar,
The linear programming approach to deterministic optimal control problems, Appl. Math., 24 (1996), 17-33.
|
| [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. |
| [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. |
| [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. |
| [25] |
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. |
| [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. |
| [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. |
| [28] |
J. Renault,
Uniform value in dynamic programming, J. European Mathematical Society, 13 (2011), 309-330.
doi: 10.4171/JEMS/254. |
| [29] |
J. E. Rubio,
Control and Optimization. The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986. |
| [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. |
| [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. |
| [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. |
| [33] |
R. Vinter,
Convex duality and nonlinear optimal control, SIAM J. Control and Optim, 31 (1993), 518-538.
doi: 10.1137/0331024. |
| [34] |
A. Zaslavski,
Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems, Springer, 2014.
doi: 10.1007/978-3-319-08034-5. |
| [35] |
A. Zaslavski,
Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006. |
| [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. |
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. |
| [2] |
R. Ash, Measure, Integration and Functional Analysis, Academic Press, 1972. |
| [3] |
J. -P. Aubin,
Viability Theory Birkhäuser, 1991. |
| [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. |
| [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. |
| [6] |
C. J. Bishop, E. 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. |
| [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.
|
| [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. |
| [9] |
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. |
| [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. |
| [11] |
N. Dunford and J. T. Schwartz,
Linear Operators, Part I, General Theory, Wiley & Sons, Inc. , New York, 1988. |
| [12] |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [21] |
D. Hernandez-Hernandez, O. Hernandez-Lerma and M. Taksar,
The linear programming approach to deterministic optimal control problems, Appl. Math., 24 (1996), 17-33.
|
| [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. |
| [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. |
| [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. |
| [25] |
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. |
| [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. |
| [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. |
| [28] |
J. Renault,
Uniform value in dynamic programming, J. European Mathematical Society, 13 (2011), 309-330.
doi: 10.4171/JEMS/254. |
| [29] |
J. E. Rubio,
Control and Optimization. The Linear Treatment of Nonlinear Problems, Manchester University Press, Manchester, 1986. |
| [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. |
| [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. |
| [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. |
| [33] |
R. Vinter,
Convex duality and nonlinear optimal control, SIAM J. Control and Optim, 31 (1993), 518-538.
doi: 10.1137/0331024. |
| [34] |
A. Zaslavski,
Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems, Springer, 2014.
doi: 10.1007/978-3-319-08034-5. |
| [35] |
A. Zaslavski,
Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006. |
| [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. |
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