A 𝒱𝒰-calculus for composite functions and the 𝒰-Hessian of partly smooth functions
Abstract
We assemble a finite-dimensional 𝒱𝒰-calculus for composite nonsmooth functions whose outer function is convex: the chain rule, separable and convex sums, a strictly differentiable perturbation of a lower semicontinuous (lsc) term, the model δX+f0+θ∘ F, and a finite maximum of C1 functions. The same algebra yields an ε-𝒱𝒰 chain rule for a proper outer approximation of the subdifferential. On the set ℛh,F of points at which the convex chain rule holds, the subspaces 𝒱f and 𝒰f, an orthonormal frame of 𝒰f, and the 𝒰-gradient g̅u are written in terms of the factors. If f is C1-partly smooth at x̅, that gradient is g̅u=Uf⊤∇ℳf(x̅). If f is C2-partly smooth and 0∈ri∂ f(x̅), the convex 𝒰-Hessian HU (when f is convex) or the local matrix Hε (when f is prox-regular at x̅ for 0) equals the Gram matrix Uf⊤∇2ℳf(x̅) Uf; the same matrix, written along the active manifold in a continuous frame, depends continuously on the base point. If in addition f is C2-partly smooth at x̅, then under prox-regularity and subdifferential continuity at x̅ for 0, Hε≻ 0 is equivalent to tilt stability of x̅ and to strong metric regularity of ∂ f at (x̅,0), with lip((∂ f)−1)(0|x̅)=|Hε−1|. The calculus and the test are illustrated on a hinge composite and two elementary tests of HU≻ 0.
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BibTeX
@misc{liu2026𝒱𝒰calculus,
title = {{A $\mathcalVU$-calculus for composite functions and the $\mathcalU$-Hessian of partly smooth functions}},
author = {Shuai Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18683},
eprint = {2609.18683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18683},
}
RIS
TY - UNPB
TI - A 𝒱𝒰-calculus for composite functions and the 𝒰-Hessian of partly smooth functions
AU - Shuai Liu
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18683
UR - https://arxiv.org/abs/2609.18683
AB - We assemble a finite-dimensional 𝒱𝒰-calculus for composite nonsmooth functions whose outer function is convex: the chain rule, separable and convex sums, a strictly differentiable perturbation of a lower semicontinuous (lsc) term, the model δ_X+f_0+θ∘ F, and a finite maximum of C^1 functions. The same algebra yields an ε-𝒱𝒰 chain rule for a proper outer approximation of the subdifferential. On the set ℛ_h,F of points at which the convex chain rule holds, the subspaces 𝒱f and 𝒰f, an orthonormal frame of 𝒰f, and the 𝒰-gradient g̅_u are written in terms of the factors. If f is C^1-partly smooth at x̅, that gradient is g̅_u=U_f^⊤∇_ℳf(x̅). If f is C^2-partly smooth and 0∈ri∂ f(x̅), the convex 𝒰-Hessian H_U (when f is convex) or the local matrix H_ε (when f is prox-regular at x̅ for 0) equals the Gram matrix U_f^⊤∇^2_ℳf(x̅) U_f; the same matrix, written along the active manifold in a continuous frame, depends continuously on the base point. If in addition f is C^2-partly smooth at x̅, then under prox-regularity and subdifferential continuity at x̅ for 0, H_ε≻ 0 is equivalent to tilt stability of x̅ and to strong metric regularity of ∂ f at (x̅,0), with lip((∂ f)^−1)(0|x̅)=|H_ε^−1|. The calculus and the test are illustrated on a hinge composite and two elementary tests of H_U≻ 0.
ER -