Dmitriy Drusvyatskiy, Alexander D. Ioffe, Adrian S. Lewis
Examples exist of extended-real-valued closed functions on ${\bf R}^n$ whose subdifferentials (in the standard, limiting sense) have large graphs. By contrast, if such a function is semi-algebraic, then its subdifferential graph must have everywhere constant local dimension $n$. This result is related to a celebrated theorem of Minty, and surprisingly may fail for the Clarke subdifferential.
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Nonminimizing Attractors of Stochastic Subgradient Methods on Semialgebraic Functions
该研究否定了Davis等人2026年的猜想，构造了二维半代数函数，证明随机次梯度方法可收敛到非极小Clarke临界点，为相关算法收敛性分析提供了反例。
Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on $\R^n$ have a limiting subdifferential graph of uniform local dimension $n$.
Directional Subdifferentials of the Value Function in Asplund Spaces
本文在Asplund空间中建立变分框架，拓展方向非光滑分析至无限维空间，推导值函数的方向次微分上估计，为无限维层级系统灵敏度分析提供精细分析基础。
This work provides calculus for the Fréchet and limiting subdifferential of the pointwise supremum given by an arbitrary family of lower semicontinuous functions. We start our study showing fuzzy results about the Fréchet subdifferential of the supremum function. Posteriorly, we study in finite- and infinite-dimensional settings the limiting subdifferential of the supremum function. Finally, we apply our results to the study of the convex subdifferential; here we recover general formulae for the subdifferential of an arbitrary family of convex functions.
We prove that the subdifferential of any semi-algebraic extended-real-valued function on $\R^n$ has $n$-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems.
The counting function on binary values is extended to the signed case in order to count the number of transitions between contiguous locations. A generalized subdifferential for the sign change counting function is given where classical subdifferentials remain intractable. An attempt to prove global optimality at some point, for the 4-dimensional first non trivial example, is made by using a sufficient condition specially tailored among all the cases for this subdifferential.
Probability functions figure prominently in optimization problems of engineering. They may be nonsmooth even if all input data are smooth.This fact motivates the consideration of subdifferentials for such typically just continuous functions. The aim of this paper is to provide subdifferential formulae in the case of Gaussian distributions for possibly infinite-dimensional decision variables and nonsmooth (locally Lipschitzian) input data. These formulae are based on the spheric-radial decomposition of Gaussian random vectors on the one hand and on a cone of directions of moderate growth on the…
The Minimal Robust Core of Abstract Subdifferentials
本文引入度量依赖的核次微分，一种在可度量化拓扑向量空间上扩展实值函数的局部鲁棒仿射支撑构造，并证明其在完备空间和Banach空间上的极小性。
We provide formulae for the $\varepsilon$-subdifferential of the integral function $ I_f(x):=\int_T f(t,x) dμ(t), $ where the integrand $f:T\times X \to [-\infty,+\infty]$ is measurable in $(t,x)$ and convex in $x$. The state variable lies in a locally convex space, possibly non-separable, while $T$ is given a structure of a nonnegative complete $σ$-finite measure space $(T,\mathcal{A},μ)$. The resulting characterizations are given in terms of the $ε$-subdifferential of the data functions involved in the integrand, $f$, without requiring any qualification conditions. We also derive new formula…
The aim of this work is to provide formulae for the subdifferential and the conjungate function of the supremun function over an arbitrary family of functions. The work is principally motivated by the case when data functions are lower semicontinuous proper and convex. Nevertheless, we explore the case when the family of functions is arbitrary, but satisfying that the biconjugate of the supremum functions is equal to the supremum of the biconjugate of the data functions. The study focuses its attention on functions defined in finite-dimensional spaces, in this case the formulae can be simplifi…
The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of …
In this paper, we first study nonsmooth steepest descent method for nonsmooth functions defined on Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitzian continuity. As one application, the established algorithm is used to search the minimizer of lower semicontinuous and convex functions on finite-dimensional space. The convergent theorem, as one extension and improvement of the existing converging result for twice continuously differentiable convex f…
In this paper, we study the problem of computing by relaxation hierarchies the infimum of a real polynomial function f on a closed basic semialgebraic set and the points where this infimum is reached, if they exist. We show that when the infimum is reached, a relaxation hierarchy constructed from the Karush-Kuhn-Tucker ideal is always exact and that the vanishing ideal of the KKT minimizer points is generated by the kernel of the associated moment matrix in that degree, even if this ideal is not zero-dimensional. We also show that this relaxation allows to detect when there is no KKT minimizer…
On convergence rates of subgradient descent on semialgebraic functions
本文研究了子梯度方法在非光滑非凸函数上的收敛速率，通过几何条件证明了在特定分层结构下，子梯度方法能局部逼近黎曼梯度下降，从而测量站性，并推导出基于分层结构的收敛速率。
A slope generalization of Attouch theorem
本文在有限维及温和有界假设下，用斜率替代次微分推广了Attouch定理，证明函数上图收敛等价于其斜率上图收敛。
Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is stratified by the location of its ten nodes. There are twenty maximal strata, identified recently by Degtyarev and Itenberg, via the global Torelli Theorem for real K3 surfaces. We here give a new proof that is self-contained and algorithmic. This involves extending Cayley's characterization of quartic symmetroids, by the property that the branch locus of the projection from a node consists of two cubic curves. This paper represents a first step towards the classification of all spectrahedra of a given degree…
We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini--Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler--Lagrange condition and the maximum principle.
All convex bodies are in the subdifferential of some everywhere differentiable locally Lipschitz function
本研究构造了$\u211d^N$上的可微局部Lipschitz函数，其次微分极限集可对应任意凸体，改进后可覆盖所有内部非空紧连通子集，且这类病态函数含无限维向量空间并在局部Lipschitz函数空间中一致稠密。
In this note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward and yields the desired results under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity assumptions on the functions in play. The versatility of our findings is illustrated by a series of applications in optimization and matrix analysis, including conic programming, matrix-fractional, variational Gram, and spectral functions.
In this paper, we revisit the Mordukhovich's subdifferential criterion for Lipschitz continuity of nonsmooth functions and coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria are useful and beautiful results in modern variational analysis showing the state of the art of the field. As an application, we establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and the scalarization function, that play an important role in many aspects of nonsmooth analysis and optimization.
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