Approximate Solutions of Variational Inequalities and the Ekeland Principle

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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Bibliographic Details
Main Authors: Raffaele Chiappinelli, David E. Edmunds
Format: Article
Language:English
Published: MDPI AG 2025-03-01
Series:Mathematics
Subjects:
coercive operators and functionals
strongly monotone operator
minimization on convex sets
pseudo-monotone operator
Online Access:https://www.mdpi.com/2227-7390/13/6/1016
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Summary: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.
ISSN:2227-7390

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