Near Optimality of Lipschitz and Smooth Policies in Controlled Diffusions Article Swipe
Somnath Pradhan
,
Serdar Yüksel
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2405.16223
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2405.16223
For optimal control of diffusions under several criteria, due to computational or analytical reasons, many studies have a apriori assumed control policies to be Lipschitz or smooth, often with no rigorous analysis on whether this restriction entails loss. While optimality of Markov/stationary Markov policies for expected finite horizon/infinite horizon (discounted/ergodic) cost and cost-up-to-exit time optimal control problems can be established under certain technical conditions, an optimal solution is typically only measurable in the state (and time, if the horizon is finite) with no apriori additional structural properties. In this paper, building on our recent work [S. Pradhan and S. Yüksel, Continuity of cost in Borkar control topology and implications on discrete space and time approximations for controlled diffusions under several criteria, Electronic Journal of Probability (2024)] establishing the regularity of optimal cost on the space of control policies under the Borkar control topology for a general class of diffusions, we establish near optimality of smooth/Lipschitz continuous policies for optimal control under expected finite horizon, infinite horizon discounted/average, and up-to-exit time cost criteria. Under mild assumptions, we first show that smooth/Lipschitz continuous policies are dense in the space of Markov/stationary Markov policies under the Borkar topology. Then utilizing the continuity of optimal costs as a function of policies on the space of Markov/stationary policies under the Borkar topology, we establish that optimal policies can be approximated by smooth/Lipschitz continuous policies with arbitrary precision. While our results are extensions of our recent work, the practical significance of an explicit statement and accessible presentation dedicated to Lipschitz and smooth policies, given their prominence in the literature, motivates our current paper.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2405.16223
- https://arxiv.org/pdf/2405.16223
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4399115399
All OpenAlex metadata
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https://openalex.org/W4399115399Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2405.16223Digital Object Identifier
- Title
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Near Optimality of Lipschitz and Smooth Policies in Controlled DiffusionsWork title
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preprintOpenAlex work type
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enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-05-25Full publication date if available
- Authors
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Somnath Pradhan, Serdar YükselList of authors in order
- Landing page
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https://arxiv.org/abs/2405.16223Publisher landing page
- PDF URL
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https://arxiv.org/pdf/2405.16223Direct link to full text PDF
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2405.16223Direct OA link when available
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Lipschitz continuity, Mathematics, Economics, Mathematical economics, Mathematical optimization, Computer science, Pure mathematicsTop concepts (fields/topics) attached by OpenAlex
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0Total citation count in OpenAlex
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10Other works algorithmically related by OpenAlex
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