Approximate Solutions of Variational Inequalities and the Ekeland Principle
Let <i>K</i> be a closed convex subset of a real Banach space <i>X</i>, and let <i>F</i> be a map from <i>X</i> to its dual <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics&...
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| author | Raffaele Chiappinelli David E. Edmunds |
| author_facet | Raffaele Chiappinelli David E. Edmunds |
| author_sort | Raffaele Chiappinelli |
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| description | Let <i>K</i> be a closed convex subset of a real Banach space <i>X</i>, and let <i>F</i> be a map from <i>X</i> to its dual <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>X</mi><mo>*</mo></msup></semantics></math></inline-formula>. We study the so-called variational inequality problem: given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>∈</mo><msup><mi>X</mi><mrow><mo>*</mo><mo>,</mo></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> does there exist <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>∈</mo><mi>K</mi></mrow></semantics></math></inline-formula> such that (in standard notation) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="⟨" close="⟩"><mi>F</mi><mrow><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>−</mo><mi>y</mi><mo>,</mo><mspace width="3.33333pt"></mspace><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mfenced><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>K</mi><mo>?</mo></mrow></semantics></math></inline-formula> After a short exposition of work in this area, we establish conditions on <i>F</i> sufficient to ensure a positive answer to the question of whether <i>F</i> is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle. |
| format | Article |
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| spelling | doaj-art-061e329ee3ad45c4972f2be089c859a92025-08-20T01:49:04ZengMDPI AGMathematics2227-73902025-03-01136101610.3390/math13061016Approximate Solutions of Variational Inequalities and the Ekeland PrincipleRaffaele Chiappinelli0David E. Edmunds1Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, I-53100 Siena, ItalyDepartment of Mathematics, University of Sussex, Brighton BN1 9QH, UKLet <i>K</i> be a closed convex subset of a real Banach space <i>X</i>, and let <i>F</i> be a map from <i>X</i> to its dual <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>X</mi><mo>*</mo></msup></semantics></math></inline-formula>. We study the so-called variational inequality problem: given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>∈</mo><msup><mi>X</mi><mrow><mo>*</mo><mo>,</mo></mrow></msup><mo>,</mo></mrow></semantics></math></inline-formula> does there exist <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>∈</mo><mi>K</mi></mrow></semantics></math></inline-formula> such that (in standard notation) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="⟨" close="⟩"><mi>F</mi><mrow><mo>(</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></mrow><mo>−</mo><mi>y</mi><mo>,</mo><mspace width="3.33333pt"></mspace><mi>x</mi><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mfenced><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>K</mi><mo>?</mo></mrow></semantics></math></inline-formula> After a short exposition of work in this area, we establish conditions on <i>F</i> sufficient to ensure a positive answer to the question of whether <i>F</i> is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle.https://www.mdpi.com/2227-7390/13/6/1016coercive operators and functionalsstrongly monotone operatorminimization on convex setspseudo-monotone operator |
| spellingShingle | Raffaele Chiappinelli David E. Edmunds Approximate Solutions of Variational Inequalities and the Ekeland Principle Mathematics coercive operators and functionals strongly monotone operator minimization on convex sets pseudo-monotone operator |
| title | Approximate Solutions of Variational Inequalities and the Ekeland Principle |
| title_full | Approximate Solutions of Variational Inequalities and the Ekeland Principle |
| title_fullStr | Approximate Solutions of Variational Inequalities and the Ekeland Principle |
| title_full_unstemmed | Approximate Solutions of Variational Inequalities and the Ekeland Principle |
| title_short | Approximate Solutions of Variational Inequalities and the Ekeland Principle |
| title_sort | approximate solutions of variational inequalities and the ekeland principle |
| topic | coercive operators and functionals strongly monotone operator minimization on convex sets pseudo-monotone operator |
| url | https://www.mdpi.com/2227-7390/13/6/1016 |
| work_keys_str_mv | AT raffaelechiappinelli approximatesolutionsofvariationalinequalitiesandtheekelandprinciple AT davideedmunds approximatesolutionsofvariationalinequalitiesandtheekelandprinciple |