Occupation Measure Relaxations in Variational Problems: The Role of Convexity

Kód výsledku v 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>

Nalezeny alternativní kódy

RIV/68407700:21110/24:00377627 RIV/68407700:21230/24:00377627

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Occupation Measure Relaxations in Variational Problems: The Role of Convexity

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Occupation Measure Relaxations in Variational Problems: The Role of Convexity

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SIAM Journal on Optimization

ISSN

1052-6234

e-ISSN

1095-7189

Svazek periodika

34

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

22

Strana od-do

1708-1731

Kód UT WoS článku

001228311100007

EID výsledku v databázi Scopus

2-s2.0-85194146024