Mikhail Krastanov, Michael Malisoff, Peter Wolenski
We provide a new sufficient condition for strong invariance for differential inclusions, under very general conditions on the dynamics, in terms of a Hamiltonian inequality. In lieu of the usual Lipschitzness assumption on the multifunction, we assume a feedback realization condition that can in particular be satisfied for measurable dynamics that are neither upper nor lower semicontinuous.
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We provide new infinitesimal characterizations for strong invariance of multifunctions in terms of Hamiltonian inequalities and tangent cones. In lieu of the standard local Lipschitzness assumption on the multifunction, we assume a new feedback realizability condition that can in particular be satisfied by control systems that are discontinuous in the state variable. Our realization condition is based on H. Sussmann's unique limiting property, and allows a more general class of feedback realizations than is allowed by the recent strong invariance characterizations of Krastanov, Malisoff, and W…
Relative Lipschitz-like property of parametric systems via projectional coderivative
本文研究参数系统解映射的相对Lipschitz-like性质，通过投影余导数上估计给出仿射变分不等式和线性互补问题的充分或充要条件，并证明多面体多值函数中该性质与局部内半连续性的等价性。
We give sufficient conditions for Lipschitz-likeness of a class of polyhedral set-valued mappings in Hilbert spaces based on Relaxed Constant Rank Constraint Qualification (RCRCQ) proposed recently by Minchenko and Stakhovsky. To this aim we prove the R-regularity of the considered set-valued mapping and correct the respective proof given by these authors.
In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable ge…
Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka-Łojasiewicz Property without Global Lipschitz Assumptions
针对复合优化问题中近端梯度法收敛分析依赖强全局Lipschitz假设的局限，本文证明仅需局部Lipschitz假设结合Kurdyka-Łojasiewicz性质，即可得到全局收敛与收敛速率结果，放宽了算法收敛的前提条件。
This paper studies conditions for invariance of dynamical systems on stratified do- mains as originally introduced by Bressan and Hong. We establish Hamiltonian conditions for both weak and strong invariance of trajectories on systems with non-Lipschitz data. This is done via the identification of a new multifunction, the essential velocity multifunction. Properties of this multifunction are investigated and used to establish the relevant invariance criteria.
Convergence analysis of nonmonotone proximal gradient methods under local Lipschitz continuity and Kurdyka--Łojasiewicz property
本文研究了在局部Lipschitz连续性和KL性质下，结合平均与最大线搜索的非单调近端梯度方法，通过索引划分在不要求迭代有界的情况下证明了其全局收敛性及收敛速率，并得出该收敛性独立于索引划分选择的结论。
Convergence of Nonmonotone Proximal Gradient Methods under the Kurdyka-Lojasiewicz Property without a Global Lipschitz Assumption
本文首次证明在Kurdyka-Łojasiewicz性质下，无需全局Lipschitz假设和迭代有界性先验知识，非单调近端梯度方法具有与单调方法相同的全局收敛性与收敛速率。
Algorithms for solving variational inequalities and saddle point problems with some generalizations of Lipschitz property for operators
本文研究求解变分不等式与强凸-凹鞍点问题，通过加速方法改进算法复杂度估计，并在任意近端结构下引入算子有界性条件类比，提出最优的镜像下降方法。
The paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool one employs the directional limiting coderivative which, together with the graphical derivative, enable us a fine analysis of the local behavior of the investigated multifunction along relevant directions. For verification of the calmness property, in addition, a new condition has been discovered which parallels the missing implicit function paradigm and permits us to replace the original multifunction by a substantially simpler one. Moreove…
Inspired by shape constrained estimation under general nonnegative derivative constraints, this paper considers the B-spline approximation of constrained functions and studies the asymptotic performance of the constrained B-spline estimator. By invoking a deep result in B-spline theory (known as de Boor's conjecture) first proved by A. Shardin as well as other new analytic techniques, we establish a critical uniform Lipschitz property of the B-spline estimator subject to arbitrary nonnegative derivative constraints under the $\ell_\infty$-norm with possibly non-equally spaced design points and…
Motivated by the result of invariance of regular-boundary open sets in \cite{CannarsaDaPratoFrankowska2009} and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems). Comments on optimality are provided via examples. Second, we provide a border avoidance (near-viability) counterpart of \cite{CannarsaDaPratoFrankowska2009} for controlled Brownian diffusions and piecewise deterministic switched Markov processes (PDsMP). We equally di…
We study the problem of algorithmically optimizing the Hamiltonian $H_N$ of a spherical or Ising mixed $p$-spin glass. The maximum asymptotic value $\mathsf{OPT}$ of $H_N/N$ is characterized by a variational principle known as the Parisi formula, proved first by Talagrand and in more generality by Panchenko. Recently developed approximate message passing algorithms efficiently optimize $H_N/N$ up to a value $\mathsf{ALG}$ given by an extended Parisi formula, which minimizes over a larger space of functional order parameters. These two objectives are equal for spin glasses exhibiting a no overl…
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