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<p>This paper introduces and considers the concept of generalized subsmoothness of a multifunction, which is a generalization of both the prox-regularity property and the subsmoothness property of multifunctions. Subsequently, it mainly deals with generalized metric subregularity (in particular, Hölder metric subregularity) for general set-valued mappings in Asplund spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive sufficient conditions for generalized metric subregularity, which extend even the known results for the conventional metric subregularity. In particular, our results improve/extend the main results established by Li and Mordukhovich (SIAM J. Optim. 22:1655–1684, <CitationRef CitationID="CR13">2012</CitationRef>). Moreover, we also conduct local convergence analysis of an inexact quasi-Newton method for solving the generalized equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_753_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>0</mn> <mo>∈</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$0\in f(x)+F(x)$</EquationSource> </InlineEquation> in Banach spaces, where the function <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_753_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> <EquationSource Format="TEX">$f$</EquationSource> </InlineEquation> is continuous but not smooth and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_753_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>F</mi> </math></EquationSource> <EquationSource Format="TEX">$F$</EquationSource> </InlineEquation> is a set-valued mapping with closed graph.</p>

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Generalized Metric Subregularity for Generalized Subsmooth Multifunctions in Asplund Spaces

  • 期刊论文

  • 发表日期：

    2025年4月11日

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  • Ming Gao,
  • Wei Ouyang,
  • Jin Zhang,
  • Jiangxing Zhu

摘要

This paper introduces and considers the concept of generalized subsmoothness of a multifunction, which is a generalization of both the prox-regularity property and the subsmoothness property of multifunctions. Subsequently, it mainly deals with generalized metric subregularity (in particular, Hölder metric subregularity) for general set-valued mappings in Asplund spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive sufficient conditions for generalized metric subregularity, which extend even the known results for the conventional metric subregularity. In particular, our results improve/extend the main results established by Li and Mordukhovich (SIAM J. Optim. 22:1655–1684, 2012). Moreover, we also conduct local convergence analysis of an inexact quasi-Newton method for solving the generalized equation 0 ∈ f ( x ) + F ( x ) $0\in f(x)+F(x)$ in Banach spaces, where the function f $f$ is continuous but not smooth and F $F$ is a set-valued mapping with closed graph.