<p>In this paper, we first derive an exact formula for computing the regular coderivative of the metric projection operator onto closed balls <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_771_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>r</mi> <mi mathvariant="double-struck">B</mi> </math></EquationSource> <EquationSource Format="TEX">$r\mathbb{B}$</EquationSource> </InlineEquation> centered at the origin in Hilbert spaces. This result is then extended to the metric projection operator onto any closed ball <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_771_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">B</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{B}(c,r)$</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_771_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>c</mi> <mo>∈</mo> <mi>H</mi> </math></EquationSource> <EquationSource Format="TEX">$c\in H$</EquationSource> </InlineEquation> is an arbitrary center and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_771_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>r</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$r>0$</EquationSource> </InlineEquation> is the radius. We also establish a formula for computing the Mordukhovich coderivative of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11228_2025_771_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mrow> <mi mathvariant="double-struck">B</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> <EquationSource Format="TEX">$P_{\mathbb{B}(c,r)}$</EquationSource> </InlineEquation>. Finally, we present a formula for calculating the graphical derivative of the metric projection operator in this setting.</p>

Coderivative and Graphical Derivative of the Metric Projection onto Closed Balls in Hilbert Spaces

期刊论文
发表日期：

2025年9月2日

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Le Van Hien

摘要

In this paper, we first derive an exact formula for computing the regular coderivative of the metric projection operator onto closed balls $r B$ $r\mathbb{B}$ centered at the origin in Hilbert spaces. This result is then extended to the metric projection operator onto any closed ball $B (c, r)$ $\mathbb{B}(c,r)$ , where $c \in H$ $c\in H$ is an arbitrary center and $r > 0$ $r>0$ is the radius. We also establish a formula for computing the Mordukhovich coderivative of $P_{B (c, r)}$ $P_{\mathbb{B}(c,r)}$ . Finally, we present a formula for calculating the graphical derivative of the metric projection operator in this setting.