Academic Journal
Deterministic and Stochastic Frank–Wolfe Recursion on Probability Spaces.
| Τίτλος: | Deterministic and Stochastic Frank–Wolfe Recursion on Probability Spaces. |
|---|---|
| Συγγραφείς: | Yu, Di, Henderson, Shane G., Pasupathy, Raghu |
| Πηγή: | Mathematics of Operations Research (INFORMS); Aug2026, Vol. 51 Issue 3, p2593-2618, 26p |
| Θεματικοί όροι: | Probability measures, Optimization algorithms, Calculus of variations, Mathematical optimization, Monte Carlo method, Central limit theorem, Convex programming, Stochastic programming |
| Περίληψη: | Motivated by applications in emergency response and experimental design, we consider smooth stochastic optimization problems over probability measures supported on compact subsets of the Euclidean space. With the influence function as the variational object, we construct a deterministic Frank–Wolfe (dFW) recursion for probability spaces. The dFW recursion is made especially possible by a lemma that identifies the solution to the infinite-dimensional Frank–Wolfe subproblem as a Dirac measure concentrating on the minimum of the influence function at the incumbent iterate. Each iterate in the dFW recursion is thus expressed through a "particle update," as a convex combination of the incumbent iterate and a Dirac measure. To address common application contexts that have access only to Monte Carlo observations of the objective and influence function, we construct a stochastic Frank–Wolfe (sFW) variation that generates a random sequence of probability measures constructed using minima of increasingly accurate estimates of the influence function. We demonstrate that the sFW optimality gap sequence exhibits O(k−1) iteration complexity almost surely and in expectation for smooth convex objectives, and O(k−1/2) (in the Frank–Wolfe gap) for smooth nonconvex objectives. Furthermore, we show that an easy-to-implement fixed-step, fixed-sample version of the sFW method exhibits exponential convergence to ε -optimality. We end with a central limit theorem on the observed objective values at the sequence of generated random measures. To further intuition, we include several illustrative examples with exact influence function calculations. Funding: This work was partially supported by the National Science Foundation [Grants CMMI-2035086, DMS-2230023, and OAC-2410950] and by the Office of Naval Research [Grants 13000991 and N000141712295]. [ABSTRACT FROM AUTHOR] |
| Copyright of Mathematics of Operations Research (INFORMS) is the property of INFORMS: Institute for Operations Research & the Management Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Βάση Δεδομένων: | Complementary Index |
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| Items | – Name: Title Label: Title Group: Ti Data: Deterministic and Stochastic Frank–Wolfe Recursion on Probability Spaces. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Yu%2C+Di%22">Yu, Di</searchLink><br /><searchLink fieldCode="AR" term="%22Henderson%2C+Shane+G%2E%22">Henderson, Shane G.</searchLink><br /><searchLink fieldCode="AR" term="%22Pasupathy%2C+Raghu%22">Pasupathy, Raghu</searchLink> – Name: TitleSource Label: Source Group: Src Data: Mathematics of Operations Research (INFORMS); Aug2026, Vol. 51 Issue 3, p2593-2618, 26p – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Probability+measures%22">Probability measures</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Calculus+of+variations%22">Calculus of variations</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Monte+Carlo+method%22">Monte Carlo method</searchLink><br /><searchLink fieldCode="DE" term="%22Central+limit+theorem%22">Central limit theorem</searchLink><br /><searchLink fieldCode="DE" term="%22Convex+programming%22">Convex programming</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+programming%22">Stochastic programming</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Motivated by applications in emergency response and experimental design, we consider smooth stochastic optimization problems over probability measures supported on compact subsets of the Euclidean space. With the influence function as the variational object, we construct a deterministic Frank–Wolfe (dFW) recursion for probability spaces. The dFW recursion is made especially possible by a lemma that identifies the solution to the infinite-dimensional Frank–Wolfe subproblem as a Dirac measure concentrating on the minimum of the influence function at the incumbent iterate. Each iterate in the dFW recursion is thus expressed through a "particle update," as a convex combination of the incumbent iterate and a Dirac measure. To address common application contexts that have access only to Monte Carlo observations of the objective and influence function, we construct a stochastic Frank–Wolfe (sFW) variation that generates a random sequence of probability measures constructed using minima of increasingly accurate estimates of the influence function. We demonstrate that the sFW optimality gap sequence exhibits O(k−1) iteration complexity almost surely and in expectation for smooth convex objectives, and O(k−1/2) (in the Frank–Wolfe gap) for smooth nonconvex objectives. Furthermore, we show that an easy-to-implement fixed-step, fixed-sample version of the sFW method exhibits exponential convergence to ε -optimality. We end with a central limit theorem on the observed objective values at the sequence of generated random measures. To further intuition, we include several illustrative examples with exact influence function calculations. Funding: This work was partially supported by the National Science Foundation [Grants CMMI-2035086, DMS-2230023, and OAC-2410950] and by the Office of Naval Research [Grants 13000991 and N000141712295]. [ABSTRACT FROM AUTHOR] – Name: Abstract Label: Group: Ab Data: <i>Copyright of Mathematics of Operations Research (INFORMS) is the property of INFORMS: Institute for Operations Research & the Management Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1287/moor.2024.0584 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 26 StartPage: 2593 Subjects: – SubjectFull: Probability measures Type: general – SubjectFull: Optimization algorithms Type: general – SubjectFull: Calculus of variations Type: general – SubjectFull: Mathematical optimization Type: general – SubjectFull: Monte Carlo method Type: general – SubjectFull: Central limit theorem Type: general – SubjectFull: Convex programming Type: general – SubjectFull: Stochastic programming Type: general Titles: – TitleFull: Deterministic and Stochastic Frank–Wolfe Recursion on Probability Spaces. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Yu, Di – PersonEntity: Name: NameFull: Henderson, Shane G. – PersonEntity: Name: NameFull: Pasupathy, Raghu IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 08 Text: Aug2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 0364765X Numbering: – Type: volume Value: 51 – Type: issue Value: 3 Titles: – TitleFull: Mathematics of Operations Research (INFORMS) Type: main |
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