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<p>We prove sufficient and necessary conditions ensuring zero Lagrangian duality gap for Lagrangians defined with help of general perturbation functions. This kind of Lagrangians include generalized and augmented Lagrangians. To this aim, we use the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10898_2025_1477_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation>-convexity theory and we formulate our zero duality gap conditions in terms of elementary functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10898_2025_1477_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi \in \Phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>∈</mo> <mi mathvariant="normal">Φ</mi> </mrow> </math></EquationSource> </InlineEquation>. The obtained results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance.</p>

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Characterisation of zero duality gap for optimization problems in spaces without linear structure

  • 期刊论文

  • 发表日期：

    2025年3月15日

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  • Ewa Bednarczuk,
  • Monika Syga

摘要

We prove sufficient and necessary conditions ensuring zero Lagrangian duality gap for Lagrangians defined with help of general perturbation functions. This kind of Lagrangians include generalized and augmented Lagrangians. To this aim, we use the \(\Phi \) Φ -convexity theory and we formulate our zero duality gap conditions in terms of elementary functions \(\varphi \in \Phi \) φ ∈ Φ . The obtained results apply to optimization problems involving prox-bounded functions, DC functions, weakly convex functions and paraconvex functions as well as infinite-dimensional linear optimization problems, including Kantorovich duality which plays an important role in determining Wasserstein distance.