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<p>We demonstrate that the concept of strict proto-differentiability of subgradient mappings can play a similar role as smoothness of the gradient mapping of a function in the study of subgradient mappings of prox-regular functions. We then show that metric regularity and strong metric regularity are equivalent for a class of generalized equations when this condition is satisfied. For a class of composite functions, called <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="script">C</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">${\mathcal{C}}^{2}$</EquationSource> </InlineEquation>-decomposable, we argue that strict proto-differentiability can be characterized via a simple relative interior condition. Leveraging this observation, we present a characterization of the continuous differentiability of the proximal mapping for this class of functions via a certain relative interior condition. Applications to the study of strong metric regularity of the KKT system of a class of composite optimization problems are also provided.</p>

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Smoothness of Subgradient Mappings and Its Applications in Parametric Optimization

  • 期刊论文

  • 发表日期：

    2025年9月24日

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  • Nguyen T. V. Hang,
  • Ebrahim Sarabi

摘要

We demonstrate that the concept of strict proto-differentiability of subgradient mappings can play a similar role as smoothness of the gradient mapping of a function in the study of subgradient mappings of prox-regular functions. We then show that metric regularity and strong metric regularity are equivalent for a class of generalized equations when this condition is satisfied. For a class of composite functions, called C 2 ${\mathcal{C}}^{2}$ -decomposable, we argue that strict proto-differentiability can be characterized via a simple relative interior condition. Leveraging this observation, we present a characterization of the continuous differentiability of the proximal mapping for this class of functions via a certain relative interior condition. Applications to the study of strong metric regularity of the KKT system of a class of composite optimization problems are also provided.