复合函数的$\mathcal{VU}$-演算与部分光滑函数的$\mathcal{U}$-Hessian

中文摘要 AI 辅助

我们为外函数为凸的复合非光滑函数组装了一个有限维的$\mathcal{VU}$-演算：链式法则、可分和与凸和、下半连续（lsc）项的严格可微扰动、模型$\delta_X+f_0+\theta\circ F$，以及$C^1$函数的有限最大值。同样的代数产生了一个针对次微分适当外部近似的$\varepsilon$-$\mathcal{VU}$链式法则。在凸链式法则成立的集合$\mathcal{R}_{h,F}$上，子空间$\mathcal{V}f$和$\mathcal{U}f$、$\mathcal{U}f$的标准正交标架以及$\mathcal{U}$-梯度$\bar g_u$用因子表示。如果$f$在$\bar{x}$处是$C^1$-部分光滑的，则该梯度为$\bar g_u=U_f^\top\nabla_{\mathcal{M}}f(\bar{x})$。如果$f$是$C^2$-部分光滑的且$0\in\ri\partial f(\bar{x})$，则凸$\mathcal{U}$-Hessian $H_U$（当$f$为凸时）或局部矩阵$H_\varepsilon$（当$f$在$\bar{x}$处对$0$是prox-正则时）等于Gram矩阵$U_f^\top\nabla^2_{\mathcal{M}}f(\bar{x})\\,U_f$；同一矩阵沿活动流形在连续标架中写出时，连续依赖于基点。如果此外$f$在$\bar{x}$处是$C^2$-部分光滑的，则在prox-正则性和次微分连续性条件下，$H_\varepsilon\succ 0$等价于$\bar{x}$的倾斜稳定性以及$\partial f$在$(\bar{x},0)$处的强度量正则性，且$\lip\bigl((\partial f)^{-1}\bigr)(0\mid\bar{x})=\\|H_\varepsilon^{-1}\\|$。该演算和检验在铰链复合函数和两个$H_U\succ 0$的基本检验上得到了说明。

英文摘要

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 \(H_U\succ 0\).

发表机构

• School of Mathematics and Statistics, Nanfang College（南方学院数学与统计学院）

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