Occupation Measure Relaxations in Variational Problems: The Role of Convexity

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985556%3A_____%2F24%3A00586292" target="_blank" >RIV/67985556:_____/24:00586292 - isvavai.cz</a>
Alternative codes found
RIV/68407700:21110/24:00377627 RIV/68407700:21230/24:00377627
Result on the web
<a href="https://epubs.siam.org/doi/abs/10.1137/23M1557088" target="_blank" >https://epubs.siam.org/doi/abs/10.1137/23M1557088</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/23M1557088" target="_blank" >10.1137/23M1557088</a>

Result language
angličtina
Original language name
Occupation Measure Relaxations in Variational Problems: The Role of Convexity
Original language description
This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming reformulation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby considerably strengthening the existing results. Our conditions are also extended to optimal control problems. In addition, we demonstrate how these results can be applied in nonconvex settings, showing that the occupation measure relaxation is at least as strong as the convexification using the convex envelope, in doing so, we prove that a certain weakening of the occupation measure relaxation is equivalent to the convex envelope. This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem, which includes problems in magnetism and elasticity.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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