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1.
Nguyen Thanh Qui 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(5):1676-1689
Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed polyhedra in reflexive Banach spaces. Our focus point is a positive linear independence condition, which is a relaxed form of the linear independence condition employed recently by Henrion et al. (2010) [1], and Nam (2010) [3]. The formulae obtained allow us to get new results on solution stability of affine variational inequalities under linear perturbations. Thus, our paper develops some aspects of the work of Henrion et al. (2010) [1] Nam (2010) [3] Qui (in press) [12] and Yao and Yen (2009) [6] and [7]. 相似文献
2.
Nguyen Thanh Qui 《Journal of Mathematical Analysis and Applications》2011,381(1):352-364
This paper establishes an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the linearly perturbed normal cone mappings in reflexive Banach spaces. In comparison with Nam (2010) [5], Qui (in press) [8], Qui (2011) [7], Trang (2010) [9], the major advantage of our investigation is that here neither the linear independence condition nor the positively linear independence condition are used. Thus, no assumption on the normal vectors of the active constraints at the point in question is needed. Some aspects of the preceding results (Henrion, Mordukhovich and Nam (2010) [3], Nam (2009) [5], Qui (2011) [7], Yao and Yen (2009) [10], Yao and Yen (2009) [11]) are developed. 相似文献
3.
Liqun Ban 《Optimization》2016,65(1):9-34
Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed generalized polyhedra in reflexive Banach spaces. Assume in addition that the generating elements are linearly independent and some qualification condition holds, the Lipschitzian stability of the parameterized variational inequalities over the right-hand side perturbed generalized polyhedra is characterized using the initial data. 相似文献
4.
In this paper we address the problem of locating a new facility on a d-dimensional space when the distance measure (\(\ell _p\)- or polyhedral-norms) is different at each one of the sides of a given hyperplane \(\mathcal {H}\). We relate this problem with the physical phenomenon of refraction, and extend it to any finite dimensional space and different distances at each one of the sides of any hyperplane. An application to this problem is the location of a facility within or outside an urban area where different distance measures must be used. We provide a new second order cone programming formulation, based on the \(\ell _p\)-norm representation given in Blanco et al. (Comput Optim Appl 58(3):563–595, 2014) that allows to solve the problem in any finite dimensional space with second order cone or semidefinite programming tools. We also extend the problem to the case where the hyperplane is considered as a rapid transit media (a different third norm is also considered over \(\mathcal {H}\)) that allows the demand to travel, whenever it is convenient, through \(\mathcal {H}\) to reach the new facility. Extensive computational experiments run in Gurobi are reported in order to show the effectiveness of the approach. Some extensions of these models are also presented. 相似文献
5.
6.
Nguyen Mau Nam 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(7):2271-364
In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12]. 相似文献
7.
Julio C. García 《Mathematical Notes》1999,65(2):159-167
The paper suggests a constructive characterization of unbounded completely positive maps introduced earlier by Chebotarev
for the theory of quantum dynamical semigroups. We prove that such cones are generated by a positive self-adjoint “reference”
operator ΛεB(H) as follows: for any completely positive unbounded map Ф(·)εCPn*(F) these exists a completely positive normal bounded mapR(·)εCPn(H) such that ϕ(·)=ΛR(·)Λ. The class contains mappings that are unclosable sesquilinear forms.
Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 194–205, February, 1999. 相似文献
8.
The hypermetric coneH
n is the cone in the spaceR
n(n–1)/2 of all vectorsd=(d
ij)1i<jn
satisfying the hypermetric inequalities: –1ijn
z
j
z
j
d
ij
0 for all integer vectorsz inZ
n with –1in
z
i
=1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneH
n is polyhedral. 相似文献
9.
Matus Benko 《Optimization》2017,66(1):61-92
Estimating the regular normal cone to constraint systems plays an important role for the derivation of sharp necessary optimality conditions. We present two novel approaches and introduce a new stationarity concept which is stronger than M-stationarity. We apply our theory to three classes of mathematical programs frequently arising in the literature. 相似文献
10.
《Applied Mathematics Letters》2002,15(5):633-639
We introduce in this paper the notion of “full nuclear cone”, and we show that a nontrivial full nuclear cone can be associated to any normal cone in a locally convex space. We apply this notion to the study of Pareto efficiency. 相似文献
11.
We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [Combinatorica, 1995]. For
a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of ℤ∞
d
, the infinite graph with vertex set ℤ
d
and an edge (u, v) whenever ∥u − v∥∞ = 1. The growth rate of G, denoted ρ
G
, is the minimum ρ such that every ball of radius r > 1 in G contains at most r
ρ
vertices. By simple volume arguments, dim(G) = Ω(ρ
G
). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρ
G
) for every graph G.
Previously, it was unknown whether dim(G) could be bounded above by any function of ρ
G
. We show that a weaker form of Levin’s conjecture holds by proving that dim(G) = O(ρ
G
log ρ
G
) for any graph G. We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for
which dim(G) = Ω(ρ
G
log ρ
G
). For several special families of graphs (e.g., planar graphs), we salvage the strong form, showing that dim(G) = O(ρ
G
). Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces posed by Linial and independently
by Benjamini and Schramm.
Supported by NSF grant CCR-0121555 and by an NSF Graduate Research Fellowship. 相似文献
12.
13.
We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere Sn, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property K + K~N= 1 for the Gauss curvature K and the normal curvature K~N if the Gauss curvature is positive. Moreover, using this property we obtain the pinching on the intrinsic curvature and normal curvature, the pinching on the normal curvature, respectively. 相似文献
14.
Two main properties of the subgradient mapping of convex functions are transposed for quasiconvex ones. The continuity of the functionxf(x)–1f(x) on the domain where it is defined is deduced from some continuity properties of the normal coneN to the level sets of the quasiconvex functionf. We also prove that, under a pseudoconvexity-type condition, the normal coneN(x) to the set {x:f(x)f(x)} can be expressed as the convex hull of the limits of type {N(x
n)}, where {x
n} is a sequence converging tox and contained in a dense subsetD. In particular, whenf is pseudoconvex,D can be taken equal to the set of points wheref is differentiable.This research was completed while the second author was on a sabbatical leave at the University of Montreal and was supported by a NSERC grant. It has its origin in the doctoral thesis of the first author (Ref. 1), prepared under the direction of the second author.The authors are grateful to an anonymous referee and C. Zalinescu for their helpful remarks on a previous version of this paper. 相似文献
15.
16.
Qinghong Zhang 《4OR: A Quarterly Journal of Operations Research》2011,9(4):403-416
It is known that the minimal cone for the constraint system of a conic linear programming problem is a key component in obtaining
strong duality without any constraint qualification. For problems in either primal or dual form, the minimal cone can be written
down explicitly in terms of the problem data. However, due to possible lack of closure, explicit expressions for the dual
cone of the minimal cone cannot be obtained in general. In the particular case of semidefinite programming, an explicit expression
for the dual cone of the minimal cone allows for a dual program of polynomial size that satisfies strong duality. In this
paper we develop a recursive procedure to obtain the minimal cone and its dual cone. In particular, for conic problems with
so-called nice cones, we obtain explicit expressions for the cones involved in the dual recursive procedure. As an example
of this approach, the well-known duals that satisfy strong duality for semidefinite programming problems are obtained. The
relation between this approach and a facial reduction algorithm is also discussed. 相似文献
17.
18.
Shokouh Shahbeyk 《Optimization》2017,66(4):473-489
In this paper, proper minimal elements of a given nonconvex set in a real ordered Banach space are defined utilizing the limiting (Mordukhovich) normal cone. The newly defined points are called limiting proper minimal (LPM) points. It is proved that each LPM is a proper minimal in the sense of Borwein under some assumptions. The converse holds in Asplund spaces. The relation of LPM points with Benson, Henig, super and proximal proper minimal points are established. Under appropriate assumptions, it is proved that the set of robust elements is a subset of the set of LPM points, and the set of LPM points is dense in that of minimal points. Another part of the paper is devoted to scalarization-based and distance function-based characterizations of the LPM points. The paper is closed by some results about LPM solutions of a set-valued optimization problem via variational analysis tools. Clarifying examples are given in addition to the theoretical results. 相似文献
19.
H. S. Carslaw 《Mathematische Annalen》1914,75(1):133-147
20.