共查询到20条相似文献,搜索用时 109 毫秒
1.
H. Hu 《Journal of Optimization Theory and Applications》2005,126(2):287-308
In this paper, we study geometric condition measures and smoothness condition measures of closed convex sets, bounded linear regularity, and linear regularity. We show that, under certain conditions, the constant for the linear regularity of infinitely many closed convex sets can be characterized by the geometric condition measure of the intersection or by the smoothness condition measure of the intersection. We study also the bounded linear regularity and present some interesting properties of the general linear regularity problem.The author is grateful to the referees for valuable and constructive suggestions. In particular, she thanks a referee for drawing her attention to Corollary 5.14 of Ref. 3, which inspired her to derive Theorem 4.2 and Corollary 4.2 in the revision of this paper. 相似文献
2.
The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational
analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which
satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating
projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As
a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method
of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms
also converge linearly.
Research of A.S. Lewis supported in part by National Science Foundation Grant DMS-0504032.
Research of D.R. Luke supported in part by National Science Foundation Grant DMS-0712796. 相似文献
3.
KungChingCHANG JiaQuanLIU 《数学学报(英文版)》2003,19(1):1-28
The heat flow for the minimal surface under Plateau boundary condition is defined to be a parabolic variational inequality,and then the existence,uniqueness,regularity,continuous dependence on the initial data and the asymptotics are studied.It is applied as a deformation of the level sets in the critical point theory. 相似文献
4.
We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated
in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval
matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions
is more general than the other one. 相似文献
5.
In this paper we present a new regularity condition for the subdifferential sum formula of a convex function with the precomposition of another convex function with a continuous linear mapping. This condition is formulated by using the epigraphs of the conjugates of the functions involved and turns out to be weaker than the generalized interior-point regularity conditions given so far in the literature. Moreover, it provides a weak sufficient condition for Fenchel duality regarding convex optimization problems in infinite dimensional spaces. As an application, we discuss the strong conical hull intersection property (CHIP) for a finite family of closed convex sets. 相似文献
6.
In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities. 相似文献
7.
Katharine Turner 《Foundations of Computational Mathematics》2013,13(6):913-933
A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopic in terms of the absence of μ-critical points in an annular region. We reduce the problem of reconstructing a subset from a point cloud to the existence of a deformation retraction from the offset of the subset to the subset itself. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). We get an improvement on previous bounds for the case where the ambient space is Euclidean whenever μ≤0.945 (μ∈(0,1) by definition). In the process, we prove stability theorems for μ-critical points when the ambient space is a manifold. 相似文献
8.
The main aspect of the paper consists in the application of a particular theorem of separation between two sets to the image
associated with a constrained extremum problem. In the image space, the two sets are a convex cone, which depends on the constraints
(equalities or inequalities) of the given problem, and its image. In this way, a condition for the existence of a regular
saddle point (i.e., a sufficient optimality condition) is obtained. This regularity condition is compared with those existing
in the literature. 相似文献
9.
10.
Tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in Euclidean
spaces. The idea is, loosely speaking, to attach to every point of the set a family of random measures, called the -dimensional tangent measure distributions at the point, which describe asymptotically the -dimensional scenery seen by an observer zooming down towards this point. This tool has been used by Bandt [BA] and Graf [G]
to study the regularity of the local geometry of self similar sets, but in this paper we show that its scope goes much beyond
this situation and, in fact, it may be used to describe a strong regularity property possessed by every measure: We show that, for every measure on a Euclidean space and any dimension , at -almost every point, all -dimensional tangent measure distributions are Palm measures. This means that the local geometry of every dimension of general
measures can be described – like the local geometry of self similar sets – by means of a family of statistically self similar
random measures. We believe that this result reveals a wealth of new and unexpected information about the structure of such
general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations
between ordinary and average densities.
Received: 27 November 1996 / Revised version: 27 February 1998 相似文献
11.
Xiao-Diao Chen Linqiang Chen Yigang Wang Gang Xu Jun-Hai Yong Jean-Claude Paul 《Journal of Computational and Applied Mathematics》2009
A sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method. 相似文献
12.
John McCuan 《Calculus of Variations and Partial Differential Equations》1999,9(4):297-325
In this paper we consider nonparametric solutions of the volume constrained Plateau problem with respect to a convex planar
curve. Existence and regularity is obtained from standard elliptic theory, and convexity results for small volumes are obtained
as an immediate consequence. Finally, the regularity is applied to show a strong stability condition (Theorem 8) for all volumes
considered. This condition, in turn, allows us to adapt an argument of Cabré and Chanillo [CC97] which yields that any solution
enclosing a non-zero volume has a unique nondegenerate critical point.
Received March 25, 1998 / Accepted December 1, 1998 相似文献
13.
Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that: (1) cubic Hermite curves cannot be constructed by solely minimizing the strain energy; (2) by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties: (1) it is easy to compute; (2) it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included. 相似文献
14.
We prove that Collet-Eckmann condition for rational functions, which requires exponential expansion only along the critical
orbits, yields the H?lder regularity of Fatou components. This implies geometric regularity of Julia sets with non-hyperbolic
and critically-recurrent dynamics. In particular, polynomial Collet-Eckmann Julia sets are locally connected if connected,
and their Hausdorff dimension is strictly less than 2. The same is true for rational Collet-Eckmann Julia sets with at least
one non-empty fully invariant Fatou component.
Oblatum 22-III-1996 & 15-VII-1997 相似文献
15.
Reconstructing a 3D shape from sample points is a central problem faced in medical applications, reverse engineering, natural sciences, cultural heritage projects, etc. While these applications motivated intense research on 3D surface reconstruction, the problem of reconstructing more general shapes hardly received any attention. This paper develops a reconstruction algorithm changing the 3D reconstruction paradigm as follows.First, the algorithm handles general shapes, i.e. compact sets, as opposed to surfaces. Under mild assumptions on the sampling of the compact set, the reconstruction is proved to be correct in terms of homotopy type. Second, the algorithm does not output a single reconstruction but a nested sequence of plausible reconstructions. Third, the algorithm accommodates topological persistence so as to select the most stable features only. Finally, in case of reconstruction failure, it allows the identification of under-sampled areas, so as to possibly fix the sampling.These key features are illustrated by experimental results on challenging datasets, and should prove instrumental in enhancing the processing of such datasets in the aforementioned applications. 相似文献
16.
We consider two merit functions which can be used for solving the nonlinear complementarity problem via nonnegatively constrained minimization. One of the functions is the restricted implicit Lagrangian (Refs. 1–3), and the other appears to be new. We study the conditions under which a stationary point of the minimization problem is guaranteed to be a solution of the underlying complementarity problem. It appears that, for both formulations, the same regularity condition is needed. This condition is closely related to the one used in Ref. 4 for unrestricted implicit Lagrangian. Some new sufficient conditions are also given. 相似文献
17.
We continue the study of minimal singular surfaces obtained by a minimization of a weighted energy functional in the spirit
of J. Douglas’s approach to the Plateau problem. Modeling soap films spanning wire frames, a singular surface is the union
of three disk-type surfaces meeting along a curve which we call the free boundary. We obtain an a priori regularity result
concerning the real analyticity of the free boundary curve. Using the free boundary regularity of the harmonic map, we construct
local harmonic isothermal coordinates for the minimal singular surface in a neighborhood of a point on the free boundary.
Applications of the local uniformization are discussed in relation to H. Lewy’s real analytic extension of minimal surfaces. 相似文献
18.
Helmut Gfrerer 《Set-Valued and Variational Analysis》2014,22(1):79-115
This paper is devoted to the analysis of a special kind of regularity of a multifunction which we call metric pseudo-(sub)regularity, when the multifunction is not metrically (sub)regular at a given point but is metrically (sub)regular at certain points in a neighborhood with moduli possibly tending to infinity with a certain order. By using advanced techniques of generalized differentiation we derive conditions both necessary and sufficient for this property. As a byproduct we obtain a new sufficient condition for metric subregularity. Then we apply these results to multifunctions composed by a smooth mapping and a generalized polyhedral multifunction and obtain explicit formulas for this case. Finally we show how the theory can be used to obtain necessary optimality conditions when the constraint qualification condition of metric (sub)regularity is violated. 相似文献
19.
Paul Godin 《Journal of Differential Equations》2002,183(1):224-238
We give a complete discussion of the C∞ or analytic regularity of blow-up curves for Cauchy problems or some mixed problems for the Liouville equation in one space dimension. In the case of mixed problems, the regularity results depend on the boundary condition: actually, we show the existence of a sequence of boundary conditions for which the regularity of the blow-up curve is better than in the general case. 相似文献
20.
We consider the first boundary-value problem for second-order nondivergent parabolic equations with, in general, discontinuous coefficients. We study the regularity of a boundary point assuming that in a neighborhood of this point the boundary of the domain is a surface of revolution. We prove a necessary and sufficient regularity condition in terms of parabolic capacities; for the heat equation this condition coincides with Wiener's criterion. 相似文献