Optimality criteria for deterministic discrete-time infinite horizon optimization
We consider the problem of selecting an optimality criterion, when total costs diverge, in deterministic infinite horizon optimization over discrete time. Our formulation allows for both discrete and continuous state and action spaces, as well as time-varying, that is, nonstationary, data. The task...
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| Format: | Article |
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Wiley
2005-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.57 |
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| author | Irwin E. Schochetman Robert L. Smith |
| author_facet | Irwin E. Schochetman Robert L. Smith |
| author_sort | Irwin E. Schochetman |
| collection | DOAJ |
| description | We consider the problem of selecting an optimality criterion, when total costs diverge, in deterministic infinite horizon optimization over discrete time. Our formulation allows for both discrete and continuous state and action spaces, as well as time-varying, that is, nonstationary, data. The task is to choose a criterion that is neither too overselective, so that no policy is optimal, nor too underselective, so that most policies are optimal. We contrast and compare the following optimality criteria: strong, overtaking, weakly overtaking, efficient, and average. However, our focus is on the optimality criterion of efficiency. (A solution is efficient if it is optimal to each of the states through which it passes.) Under mild regularity conditions, we show that efficient solutions always exist and thus are not overselective. As to underselectivity, we provide weak state reachability conditions which assure that every efficient solution is also average optimal, thus providing a sufficient condition for average optima to exist. Our main result concerns the case where the discounted per-period costs converge to zero, while the discounted total costs diverge to infinity. Under the assumption that we can reach from any feasible state any feasible sequence of states in bounded time, we show that every efficient solution is also overtaking, thus providing a sufficient condition for overtaking optima to exist. |
| format | Article |
| id | doaj-art-29c2f9d907f74dd99f49fbf52ab7dd94 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-29c2f9d907f74dd99f49fbf52ab7dd942025-02-03T01:30:52ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-0120051578010.1155/IJMMS.2005.57Optimality criteria for deterministic discrete-time infinite horizon optimizationIrwin E. Schochetman0Robert L. Smith1Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USADepartment of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI 48109, USAWe consider the problem of selecting an optimality criterion, when total costs diverge, in deterministic infinite horizon optimization over discrete time. Our formulation allows for both discrete and continuous state and action spaces, as well as time-varying, that is, nonstationary, data. The task is to choose a criterion that is neither too overselective, so that no policy is optimal, nor too underselective, so that most policies are optimal. We contrast and compare the following optimality criteria: strong, overtaking, weakly overtaking, efficient, and average. However, our focus is on the optimality criterion of efficiency. (A solution is efficient if it is optimal to each of the states through which it passes.) Under mild regularity conditions, we show that efficient solutions always exist and thus are not overselective. As to underselectivity, we provide weak state reachability conditions which assure that every efficient solution is also average optimal, thus providing a sufficient condition for average optima to exist. Our main result concerns the case where the discounted per-period costs converge to zero, while the discounted total costs diverge to infinity. Under the assumption that we can reach from any feasible state any feasible sequence of states in bounded time, we show that every efficient solution is also overtaking, thus providing a sufficient condition for overtaking optima to exist.http://dx.doi.org/10.1155/IJMMS.2005.57 |
| spellingShingle | Irwin E. Schochetman Robert L. Smith Optimality criteria for deterministic discrete-time infinite horizon optimization International Journal of Mathematics and Mathematical Sciences |
| title | Optimality criteria for deterministic discrete-time infinite horizon optimization |
| title_full | Optimality criteria for deterministic discrete-time infinite horizon optimization |
| title_fullStr | Optimality criteria for deterministic discrete-time infinite horizon optimization |
| title_full_unstemmed | Optimality criteria for deterministic discrete-time infinite horizon optimization |
| title_short | Optimality criteria for deterministic discrete-time infinite horizon optimization |
| title_sort | optimality criteria for deterministic discrete time infinite horizon optimization |
| url | http://dx.doi.org/10.1155/IJMMS.2005.57 |
| work_keys_str_mv | AT irwineschochetman optimalitycriteriafordeterministicdiscretetimeinfinitehorizonoptimization AT robertlsmith optimalitycriteriafordeterministicdiscretetimeinfinitehorizonoptimization |