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<p>In the study of a non-convex minimization problem by Lachand-Robert and Peletier, they found that the difference between the compactly supported perturbation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(u+\epsilon h\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>+</mo> <mi>ϵ</mi> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation> of a strictly convex function <i>u</i> and the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-regularization of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(u+\epsilon h\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>+</mo> <mi>ϵ</mi> <mi>h</mi> </mrow> </math></EquationSource> </InlineEquation> is at most <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(o(\epsilon )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>o</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Here we find that this result is optimal, although they had expected a much stronger estimate.</p>

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The Distance Between the Perturbation of a Convex Function and its \(\Gamma \)-Regularization

  • 期刊论文

  • 发表日期：

    2025年11月25日

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  • Zichang Liu

摘要

In the study of a non-convex minimization problem by Lachand-Robert and Peletier, they found that the difference between the compactly supported perturbation \(u+\epsilon h\) u + ϵ h of a strictly convex function u and the \(\Gamma \) Γ -regularization of \(u+\epsilon h\) u + ϵ h is at most \(o(\epsilon )\) o ( ϵ ) . Here we find that this result is optimal, although they had expected a much stronger estimate.