{"ID":22952660,"CreatedAt":"2026-09-17T02:12:05.498442134Z","UpdatedAt":"2026-09-20T18:11:56.143995915Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18683","arxiv_id":"2609.18683","title":"A $\\mathcal{VU}$-calculus for composite functions and the $\\mathcal{U}$-Hessian of partly smooth functions","abstract":"We assemble a finite-dimensional \\(\\mathcal{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\\circ F\\), and a finite maximum of \\(C^1\\) functions. The same algebra yields an \\(\\varepsilon\\)-\\(\\mathcal{VU}\\) chain rule for a proper outer approximation of the subdifferential. On the set \\(\\mathcal{R}_{h,F}\\) of points at which the convex chain rule holds, the subspaces \\(\\mathcal{V}f\\) and \\(\\mathcal{U}f\\), an orthonormal frame of \\(\\mathcal{U}f\\), and the \\(\\mathcal{U}\\)-gradient \\(\\bar g_u\\) are written in terms of the factors. If \\(f\\) is \\(C^1\\)-partly smooth at \\(\\bar{x}\\), that gradient is \\(\\bar g_u=U_f^\\top\\nabla_{\\mathcal{M}}f(\\bar{x})\\). If \\(f\\) is \\(C^2\\)-partly smooth and \\(0\\in\\ri\\partial f(\\bar{x})\\), the convex \\(\\mathcal{U}\\)-Hessian \\(H_U\\) (when \\(f\\) is convex) or the local matrix \\(H_\\varepsilon\\) (when \\(f\\) is prox-regular at \\(\\bar{x}\\) for \\(0\\)) equals the Gram matrix \\(U_f^\\top\\nabla^2_{\\mathcal{M}}f(\\bar{x})\\,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 \\(\\bar{x}\\), then under prox-regularity and subdifferential continuity at \\(\\bar{x}\\) for \\(0\\), \\(H_\\varepsilon\\succ 0\\) is equivalent to tilt stability of \\(\\bar{x}\\) and to strong metric regularity of \\(\\partial f\\) at \\((\\bar{x},0)\\), with \\(\\lip\\bigl((\\partial f)^{-1}\\bigr)(0\\mid\\bar{x})=\\|H_\\varepsilon^{-1}\\|\\). The calculus and the test are illustrated on a hinge composite and two elementary tests of \\(H_U\\succ 0\\).","short_abstract":"We assemble a finite-dimensional \\(\\mathcal{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\\circ F\\), and a finite maximum of \\(C^1\\) functions....","url_abs":"https://arxiv.org/abs/2609.18683","url_pdf":"https://arxiv.org/pdf/2609.18683v1","authors":"[\"Shuai Liu\"]","published":"2026-09-16T13:59:33Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}