In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non-restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples
In the paper a new sufficient condition for the Aubin property to a class of parameterized variational systems is derived. In these systems the constraints depend both on the parameter as well as on the decision variable itself and they include, e.g., parameter-dependent quasi-variational inequalities and implicit complementarity problems. The result is based on a general condition ensuring the Aubin property of implicitly defined multifunctions which employs the recently introduced notion of the directional limiting coderivative. Our final condition can be verified, however, without an explic…
The paper deals with a new sharp criterion ensuring the Aubin property of solution maps to a class of parameterized variational systems. This class includes parameter-dependent variational inequalities with non-polyhedral constraint sets and also parameterized generalized equations with conic constraints. The new criterion requires computation of directional limiting coderivatives of the normal-cone mapping for the so-called critical directions. The respective formulas have the form of a second-order chain rule and extend the available calculus of directional limiting objects. The suggested pr…
In this paper, we study a new variant of Moreau's sweeping process with velocity constraint. Based on an adapted version of Moreau's catching-up algorithm, we show the well-posedness (in the sense existence and uniqueness) of this problem in a general framework. We show the equivalence between this implicit sweeping process and a quasistatic evolution variational inequality. It is well known that the variational formulations of many mechanical problems with unilateral contact and friction lead to an evolution variational inequality. As an application, we reformulate the quasistatic antiplane f…
In this paper we investigate the sensitivity analysis of parameterized nonlinear variational inequalities of second kind in a Hilbert space. The challenge of the present work is to take into account a perturbation on all the data of the problem. This requires special adjustments in the definitions of the generalized first- and second-order differentiations of the involved operators and functions. Precisely, we extend the notions, introduced and thoroughly studied by R.T. Rockafellar, of twice epi-differentiability and proto-differentiability to the case of a parameterized lower semi-continuous…
Boston University（波士顿大学）；Rice University（莱斯大学）；University of Maryland（马里兰大学）
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-$r$ positive semi-definite (PSD) matrices in $\mathbb{R}^{n \times n}$, with a sensing mechanism that satisfies restricted isometry properties (RIP). The algorithm we study is \emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization $UU^\top$, for $U \in \mathbb{R}^{n \times r}$. Surprisingly, recent work argues that the choice of $r \leq n$ is not pivotal: even setting $U …
In this work, we suggest a general viscosity implicit midpoint rule for nonexpansive mapping in the framework of Hilbert space. Further, under the certain conditions imposed on the sequence of parameters, strong convergence theorem is proved by the sequence generated by the proposed iterative scheme, which, in addition, is the unique solution of the variational inequality problem. Furthermore, we provide some applications to variational inequalities, Fredholm integral equations, and nonlinear evolution equations. The results presented in this work may be treated as an improvement, extension an…
In this paper, we leverage over-parameterization to design regularization-free algorithms for the high-dimensional single index model and provide theoretical guarantees for the induced implicit regularization phenomenon. Specifically, we study both vector and matrix single index models where the link function is nonlinear and unknown, the signal parameter is either a sparse vector or a low-rank symmetric matrix, and the response variable can be heavy-tailed. To gain a better understanding of the role played by implicit regularization without excess technicality, we assume that the distribution…
We consider the vectorial system \[ \begin{cases} Du \in \mathcal{O}(2), & \mbox{a.e. in}\;Ω, u=0, & \mbox{on} \;\partial Ω, \end{cases} \] where $Ω$ is a subset of $\R^2$, $u:Ω\to \R^2$ and $\mathcal{O}(2)$ is the orthogonal group of $\R^2$. We provide a variational method to select, among the infinitely many solutions, the ones that minimize an appropriate weighted measure of the singular set of the gradient.
The present paper contains some investigations about a uniform variant of the notion of metric hemiregularity, the latter being a less explored property obtained by weakening metric regularity. The introduction of such a quantitative stability property for set-valued mappings is motivated by applications to the penalization of constrained optimization problems, through the notion of problem calmness. As a main result, an implicit multifunction theorem for parameterized inclusion problems is established, which measures the uniform hemiregularity of the related solution mapping in terms of probl…
The paper deals with an extension of the available theory of SCD (subspace containing derivatives) mappings to mappings between spaces of different dimensions. This extension enables us to derive workable sufficient conditions for the isolated calmness of implicitly defined multifunctions around given reference points. This stability property differs substantially from isolated calmness at a point and, possibly in conjunction with the Aubin property, offers a new useful stability concept. The application area includes a broad class of parameterized generalized equations, where the respective c…
The subject of the present paper are stability properties of the solution set to set-valued inclusions. The latter are problems emerging in robust optimization and mathematical economics, which can not be cast in traditional generalized equations. The analysis here reported focuses on several quantitative forms of semicontinuity for set-valued mappings, widely investigated in variational analysis, which include, among others, calmness. Sufficient conditions for the occurrence of these properties in the case of the solution mapping to a parameterized set-valued inclusion are established. Conseq…
In the literature, there are a few researches to design some parameters in the Proximal Point Algorithm (PPA), especially for the multi-objective convex optimizations. Introducing some parameters to PPA can make it more flexible and attractive. Mainly motivated by our recent work (Bai et al., A parameterized proximal point algorithm for separable convex optimization, Optim. Lett. (2017) doi: 10.1007/s11590-017-1195-9), in this paper we develop a general parameterized PPA with a relaxation step for solving the multi-block separable structured convex programming. By making use of the variational…
A shape sensitive, variational approach for the matching of surfaces considered as thin elastic shells is investigated. The elasticity functional to be minimized takes into account two different types of nonlinear energies: a membrane energy measuring the rate of tangential distortion when deforming the reference shell into the template shell, and a bending energy measuring the bending under the deformation in terms of the change of the shape operators from the undeformed into the deformed configuration. The variational method applies to surfaces described as level sets. It is mathematically w…
We consider control constrained optimal control problems governed by parameterized stationary Maxwell's system with the Gauss's law. The parameters enter through dielectric, magnetic permeability, and charge density. Moreover, the parameter set is assumed to be compact. We discretize the electric field by a finite element method and use variational discretization concept to discretize the control. We create a reduced basis method for the optimal control problem and establish uniform convergence of the reduced order solutions to that of the original high dimensional problem provided that the sn…
