A convex cone C is salient if and only if C ∩ −C ⊆ {0}. A cone C is said to be generating if C − C = { x − y ∣ x ∈ C , y ∈ C } {\displaystyle C-C=\{x-y\mid x\in C,y\in C\}} equals the whole vector space.

In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base. In the case of line segments, the cone ...

For a cone, $x\in C$ requires $\lambda x \in C, \lambda \ge 0$. We can then replace $x,y$ with $\lambda x,\lambda \ge 0$ and $\mu y,\mu \ge 0$ that both belong to $C$ , like $\theta \lambda x+(1-\theta)\mu y \in C$ .

这篇笔记，首先我会对三种常见cone从几何角度出发做一个区分，再谈一下cone与 optimality condition 之间密不可分的联系。 今天的主角cone有三位，分别是： Tangent cone 、 linearized cone 、 feasible direction 。

3.1.1 DefinitionA cone is a nonempty subset of a vector space that is closed under multiplication by nonnegative scalars. That is, it includes the ray through any of its nonzero points. In other words, C is a cone if for all α ∈ R and x ∈ C α ⩾ 0 =⇒ αx ∈ C. A cone C is pointed if −C ∩C = {0}, that is, if it includes no lines. N.B.

凸锥（convex cone）：集合 C 为凸锥，即对于任意 x_1,x_2\in C 和 \theta_1,\theta_2\geq0 ，都有 \theta_1x_1+\theta_2x_2\in C 。 如下图所示： 该凸锥的两条边是 \theta_1x_1 和 \theta_2x_2 ，凸锥内部的点为 \theta_1x_1+\theta_2x_2 。 锥包：集合 C 的锥包是 C 中元素的所有锥组合的集合，即： \ {\theta_1x_1+\cdots+\theta_kx_k|x_i\in C,\theta_i\geq 0,i=1,\cdots,k\} ，它是包含 C 的最小的凸锥。 下面介绍几个重要的凸集：

2018年2月9日 · A proper cone is a convex cone C with vertex at 0, such that C ∩ (-C) = {0}. A slightly more specific definition of a proper cone is this entry (http://planetmath.org/ProperCone), but it requires the vector space to be topological.

2022年5月23日 · We call an asymptotic cone of a convex set $C$ the closed convex cone $C_{as}$ such that: $$\overline{\Gamma(C)}\cap(E\times\{0\})=C_{as}\times\{0\}$$ My question is: Isn't this $C_{as}=\{0\}$ ? Wouldn't it shrink to a point?

Cone [] 等价于 Cone [{{0, 0,-1}, {0, 0, 1}}]. Cone 表示一个填充的圆锥区域 ，其中 ，向量 与 正交，且 和 . Cone 可用于 Graphics3D .

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients. Definition. A subset C of a vector space V is a convex cone if αx + βy belongs to C, for any positive scalars α, β, and any x, y in C.