scholarly journals On Notations for Conic Hulls and Related Considerations on Tangent Cones

2021 ◽  
Vol 13 (3) ◽  
pp. 13
Author(s):  
Giorgio Giorgi
Keyword(s):  
Tangent Cone ◽  
Optimization Problem ◽  
Convex Sets ◽  
Convex Cones ◽  
Tangent Cones ◽  
Clarke Tangent Cone ◽  
Locally Convex

We propose two di erent notations for cones generated by a set and for convex cones generated by a set, usually denoted by a same notation. We make some remarks on the Bouligand tangent cone and on the Clarke tangent cone for star-shaped sets and for locally convex sets. We give some applications of these remarks to a di erentiable optimization problem with an abstract constraint.

The Quantificational Tangent Cones

1988 ◽  
Vol 40 (3) ◽  
pp. 666-694 ◽  
Author(s):  
Doug Ward
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Nonsmooth Analysis ◽  
Tangent Cone ◽  
Building Blocks ◽  
Tangent Cones ◽  
Hausdorff Topological Vector Space ◽  
Local Approximations ◽  
Locally Convex

Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones.Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2E × E → 2E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2E and x in E.In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)

Proximal Analysis and Boundaries of Closed Sets in Banach Space, Part I: Theory

1986 ◽  
Vol 38 (2) ◽  
pp. 431-452 ◽  
Author(s):  
J. M. Borwein ◽  
H. M. Strojwas
Keyword(s):  
Banach Space ◽  
Closed Subset ◽  
Tangent Cone ◽  
Reflexive Banach Space ◽  
Tangent Cones ◽  
Generalized Derivatives ◽  
Closed Sets ◽  
Clarke Tangent Cone ◽  
Contingent Cones ◽  
Proximal Analysis

As various types of tangent cones, generalized derivatives and subgradients prove to be a useful tool in nonsmooth optimization and nonsmooth analysis, we witness a considerable interest in analysis of their properties, relations and applications.Recently, Treiman [18] proved that the Clarke tangent cone at a point to a closed subset of a Banach space contains the limit inferior of the contingent cones to the set at neighbouring points. We provide a considerable strengthening of this result for reflexive spaces. Exploring the analogous inclusion in which the contingent cones are replaced by pseudocontingent cones we have observed that it does not hold any longer in a general Banach space, however it does in reflexive spaces. Among the several basic relations we have discovered is the following one: the Clarke tangent cone at a point to a closed subset of a reflexive Banach space is equal to the limit inferior of the weak (pseudo) contingent cones to the set at neighbouring points.

Finite dimensional locally convex cones

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5111-5116
Author(s):  
Davood Ayaseha
Keyword(s):  
Finite Dimension ◽  
Convex Cones ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Euclidean Topology ◽  
Finite Dimensional ◽  
Dimensional Cone ◽  
Locally Convex Cones ◽  
Locally Convex ◽  
Special Case

We study the locally convex cones which have finite dimension. We introduce the Euclidean convex quasiuniform structure on a finite dimensional cone. In special case of finite dimensional locally convex topological vector spaces, the symmetric topology induced by the Euclidean convex quasiuniform structure reduces to the known concept of Euclidean topology. We prove that the dual of a finite dimensional cone endowed with the Euclidean convex quasiuniform structure is identical with it?s algebraic dual.

scholarly journals Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

2019 ◽  
Vol 26 (04) ◽  
pp. 629-642
Author(s):  
Anargyros Katsabekis
Keyword(s):  
Complete Intersection ◽  
Tangent Cone ◽  
Affine Space ◽  
Intersection Property ◽  
Tangent Cones ◽  
Monomial Curves ◽  
Monomial Curve

Let C(n) be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve C(n + wv), where w > 0 is an integer and v ∈ ℕ4. In addition, we investigate the Cohen–Macaulayness of the tangent cone of C(n + wv).

scholarly journals On the Cohen-Macaulay Property for Quadratic Tangent Cones

10.37236/5793 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Dumitru I. Stamate
Keyword(s):  
Tangent Cone ◽  
Gröbner Basis ◽  
Numerical Semigroup ◽  
Tangent Cones ◽  
Grobner Basis

Let $H$ be an $n$-generated numerical semigroup such that its tangent cone $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadratic relations. We show that if $n<5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Cohen-Macaulay, and for $n=5$ we explicitly describe the semigroups $H$ such that $\operatorname{gr}_\mathfrak{m} K[H]$ is not Cohen-Macaulay. As an application we show that if the field $K$ is algebraically closed and of characteristic different from two, and $n\leq 5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Koszul if and only if (possibly after a change of coordinates) its defining ideal has a quadratic Gröbner basis.

An n + 1 Member Decomposition for Sets Whose Lnc Points Form n Convex Sets

1975 ◽  
Vol 27 (6) ◽  
pp. 1378-1383 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Convex Sets ◽  
Local Convexity ◽  
Locally Convex

Let S be a subset of Rd. A point x in 5 is a point of local convexity of S if and only if there is some neighborhood N of x such that, if y, z ∈ N ᑎ 5, then [y, z] ⊆ S. If S fails to be locally convex at some point q in S then q is called a point of local nonconvexity (lnc point) of S.

scholarly journals LIFTINGS OF A MONOMIAL CURVE

2018 ◽  
Vol 98 (2) ◽  
pp. 230-238
Author(s):  
MESUT ŞAHİN
Keyword(s):  
Tangent Cone ◽  
Tangent Cones ◽  
Monomial Curves ◽  
Original Curve ◽  
Monomial Curve

We study an operation, that we call lifting, creating nonisomorphic monomial curves from a single monomial curve. Our main result says that all but finitely many liftings of a monomial curve have Cohen–Macaulay tangent cones even if the tangent cone of the original curve is not Cohen–Macaulay. This implies that the Betti sequence of the tangent cone is eventually constant under this operation. Moreover, all liftings have Cohen–Macaulay tangent cones when the original monomial curve has a Cohen–Macaulay tangent cone. In this case, all the Betti sequences are just the Betti sequence of the original curve.

scholarly journals Completeness on locally convex cones

2014 ◽  
Vol 352 (10) ◽  
pp. 785-789 ◽  
Author(s):  
Mohammad Reza Motallebi
Keyword(s):  
Convex Cones ◽  
Locally Convex Cones ◽  
Locally Convex
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