Boundedness and Regularity Properties of Semismooth Reformulations of Variational Inequalities

The Karush-Kuhn-Tucker (KKT) system of the variational inequality problem over a set defined by inequality and equality constraints can be reformulated as a system of semismooth equations via an nonlinear complementarity problem (NCP) function. We give a sufficient condition for boundedness of the level sets of the norm function of this system of semismooth equations when the NCP function is metrically equivalent to the minimum function; and a sufficient and necessary condition when the NCP function is the minimum function. Nonsingularity properties identified by Facchinei, Fischer and Kanzow, 1998, SIAM J. Optim. 8, 850–869, for the semismooth reformulation of the variational inequality problem via the Fischer-Burmeister function, which is an irrational regular pseudo-smooth NCP function, hold for the reformulation based on other regular pseudo-smooth NCP functions. We propose a new regular pseudo-smooth NCP function, which is piecewise linear-rational and metrically equivalent to the minimum NCP function. When it is used to the generalized Newton method for solving the variational inequality problem, an auxiliary step can be added to each iteration to reduce the value of the merit function by adjusting the Lagrangian multipliers only. This work is supported by the Research Grant Council of Hong Kong This paper is dedicated to Alex Rubinov on the occasion of his 65th Birthday     

Regularity and Conditioning of Solution Mappings in Variational Analysis

Concepts of conditioning have long been important in numerical work on solving systems of equations, but in recent years attempts have been made to extend them to feasibility conditions, optimality conditions, complementarity conditions and variational inequalities, all of which can be posed as solving ‘generalized equations’ for set-valued mappings. Here, the conditioning of such generalized equations is systematically organized around four key notions: metric regularity, subregularity, strong regularity and strong subregularity. Various properties and characterizations already known for metric regularity itself are extended to strong regularity and strong subregularity, but metric subregularity, although widely considered, is shown to be too fragile to support stability results such as a radius of good behavior modeled on the Eckart–Young theorem.     

On Some Properties of Generalized Proximal Point Methods for Variational Inequalities

We discuss here generalized proximal point methods applied to variational inequality problems. These methods differ from the classical point method in that a so-called Bregman distance substitutes for the Euclidean distance and forces the sequence generated by the algorithm to remain in the interior of the feasible region, assumed to be nonempty. We consider here the case in which this region is a polyhedron (which includes linear and nonlinear programming, monotone linear complementarity problems, and also certain nonlinear complementarity problems), and present two alternatives to deal with linear equality constraints. We prove that the sequences generated by any of these alternatives, which in general are different, converge to the same point, namely the solution of the problem which is closest, in the sense of the Bregman distance, to the initial iterate, for a certain class of operators. This class consists essentially of point-to-point and differentiable operators such that their Jacobian matrices are positive semidefinite (not necessarily symmetric) and their kernels are constant in the feasible region and invariant through symmetrization. For these operators, the solution set of the problem is also a polyhedron. Thus, we extend a previous similar result which covered only linear operators with symmetric and positive-semidefinite matrices.     

On Some Discrete Variational Problems

Making use of a discrete version of the P.-L. Lions concentration-compactness principle, we establish some results on the existence of nontrivial solutions for nonlinear stationary discrete equations of the Schrödinger type.     

On Some Noncoercive Variational Inequalities

We study existence and regularity of solutions of noncoercive variational inequalities.     

On Some Properties of Extremals in a Variational Problem Generated by the Sobolev Embedding Theorem

Some properties of extremal functions in the inequality are studied. Bibliography: 10 titles.     

一类四阶变分不等式解的极限正则性

An example of a fourth order variational inequality is structed in this note. The example shows; u∈H^3+σ(Ω) with every σ<0.5(but u(?) H^3.5(Ω)) is limiting regularity of solution u of a fourth order variational inequality with displacement obstacle.     

On Variational Measures Related to Some Bases

We extend, to a certain class of differentiation bases, some results on the variational measure and the δ-variation obtained earlier for the full interval basis. In particular the theorem stating that the variational measure generated by an interval function is σ-finite whenever it is absolutely continuous with respect to the Lebesgue measure is extended to any Busemann–Feller basis.     

The Regularity for Solutions of Variational Inequalities

The regularity for solutions of elliptic equations is rather perfectly solved. But it does not so perfect for that of elliptic variational inequalities. In literature only different special situations are considered. Now the boundedness, C^{0,λ} continuity and C^{1,α} regularity are proved for solutions of one-sided obstacle problems under more general structural conditions, in which the growth orders of u are permitted to reach the critical exponents and the growth order ϒ of the gradient in D is permitted to be super critical as 1 < p < n.     

Regularity Properties of Some Stochastic Volterra Integrals with Singular Kernel

We prove the Hölder continuity of some stochastic Volterra integrals, with singular kernels, under integrability assumptions on the integrand. Some applications to processes arising in the analysis of the fractional Brownian motion are given. The main tool is the embedding of some Besov spaces into some sets of Hölder continuous functions.     

On Hyperbolic Variational Inequalities of First Order and Some Applications

This paper presents further developments in the study of strong solutions and their coincidence sets to the obstacle problem for linear transport operators. Under natural mild assumptions, the strong convergence of the solutions and the characteristic functions of their coincidence sets is obtained in the passage of second order to first order problems. An application to a steady-state chemotaxis problem is given. The extension to the two obstacles problem of first order is also presented, having in view an application to a porous media model in presence of saturation arising in petroleum engineering.     

On Hyperbolic Variational Inequalities of First Order and Some Applications

This paper presents further developments in the study of strong solutions and their coincidence sets to the obstacle problem for linear transport operators. Under natural mild assumptions, the strong convergence of the solutions and the characteristic functions of their coincidence sets is obtained in the passage of second order to first order problems. An application to a steady-state chemotaxis problem is given. The extension to the two obstacles problem of first order is also presented, having in view an application to a porous media model in presence of saturation arising in petroleum engineering.     

Lipschitz Regularity in Some Geometric Problems

The Lipschitz regularity is perhaps the most natural, and surely the most geometrical among all the types of regularities. For example, the Lipschitz character of an ordinary differential equation (vector field) is the natural classical sufficient condition for the (unique) integrability of this equation. The goal here is to show that, in some sense, the Lipschitz regularity is also necessary, if one assumes (geometric) individual conditions on the trajectories. In other words, we show that tangential rigidity leads to a transversal regularity.     

On Some Properties of $c$-Frames

In this paper we discuss about $c$-frames, namely continuous frames. Since, $c$-frames are generalizations of discrete frames, we generalize some results of discrete frames to continuous version. We explain some results about relations of projections in Hilbert spaces and $c$-frames to characterize these frames. Also, we will specify (precisely) the synthesis and frame operators of Bochner integrable $c$-frames. Finally, we classify Hilbert-Schmidt operators by $c$-frames and express some new identities for Parseval $c$-frames.     

On Some Properties of Quasiplanes

For quasiconformal images of a hyperplane in Euclidean space, we prove two new properties formulated in terms of the isoperimetry constant and the fundamental frequency.     

Some Properties On Isoclinism in n-Lie Algebras

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.     

A Simple Partial Regularity Proof for Minimizers of Variational Integrals

We consider multi-dimensional variational integrals

Some Regularity Results In Ferromagnetism

The magnetism of a rigid ferromagnet occupying a spatial region Ωis described by a unit vectorfield m on Ω. The total energy of m involves several terms: the anisotropy energy imposed by the lattice structure of the material, the exchange energy discouraging very rapid local changes in m , the applied energy due to external magnetic sources, and the induced magnetic field energy. Here, while incorporating all energy terms, we show that minimizers have at most isolated singularities, usually in the interior of Ω, and that there is nice asymptotic behavior at such singularities. In contrast to related harmonic map problems, the field energy is a nonlocal term, involving a solution of Maxwell's equations with coefficients depending on m