anti-cone
Let $X$ be a real vector space, and $\mathrm{\Phi}$ be a subspace^{} of linear functionals^{} on $X$.
For any set $S\subseteq X$, its anti-cone ${S}^{+}$, with respect to $\mathrm{\Phi}$, is the set
$${S}^{+}=\{\varphi \in \mathrm{\Phi}:\varphi (x)\ge 0,\text{for all}x\in S\}.$$ |
The anti-cone is also called the dual cone.
Usage
The anti-cone operation^{} is generally applied to subsets of $X$ that are themselves cones. Recall that a cone in a real vector space generalize the notion of linear inequalities in a finite number of real variables. The dual cone provides a natural way to transfer such inequalities^{} in the original vector space^{} to its dual space^{}. The concept is useful in the theory of duality.
The set $\mathrm{\Phi}$ in the definition may be taken to be any subspace of the algebraic dual space ${X}^{*}$. The set $\mathrm{\Phi}$ often needs to be restricted to a subspace smaller than ${X}^{*}$, or even the continuous dual space ${X}^{\prime}$, in order to obtain the nice closure and reflexivity^{} properties below.
Basic properties
Property 1.
The anti-cone is a convex cone in $\mathrm{\Phi}$.
Proof.
If $\varphi (x)$ is non-negative, then so is $t\varphi (x)$ for $t>0$. And if ${\varphi}_{1}(x),{\varphi}_{2}(x)\ge 0$, then clearly $(1-t){\varphi}_{1}(x)+t{\varphi}_{2}(x)\ge 0$ for $0\le t\le 1$. ∎
Property 2.
If $K\mathrm{\subseteq}X$ is a cone, then its anti-cone ${K}^{\mathrm{+}}$ may be equivalently characterized as:
$${K}^{+}=\{\varphi \in \mathrm{\Phi}:\varphi (x)\mathit{\text{over}}x\in K\mathit{\text{is bounded below}}\}.$$ |
Proof.
It suffices to show that if ${inf}_{x\in K}\varphi (x)$ is bounded below, then it is non-negative. If it were negative, take some $x\in K$ such that $$. For any $t>0$, the vector $tx$ is in the cone $K$, and the function value $\varphi (tx)=t\varphi (x)$ would be arbitrarily large negative, and hence unbounded^{} below. ∎
Topological properties
Assumptions^{}. Assume that $\mathrm{\Phi}$ separates points of $X$. Let $X$ have the weak topology generated by $\mathrm{\Phi}$, and let $\mathrm{\Phi}$ have the weak-* topology generated by $X$; this makes $X$ and $\mathrm{\Phi}$ into Hausdorff topological vector spaces^{}.
Vectors $x\in X$ will be identified with their images $\widehat{x}$ under the natural embedding of $X$ in its double dual space.
The pairing $(X,\mathrm{\Phi})$ is sometimes called a dual pair; and $(\mathrm{\Phi},X)$, where $X$ is identified with its image in the double dual, is also a dual pair.
Property 3.
${S}^{+}$ is weak-* closed.
Proof.
Let $\{{\varphi}_{\alpha}\}\subseteq \mathrm{\Phi}$ be a net converging to $\varphi $ in the weak-* topology. By definition, $\widehat{x}({\varphi}_{\alpha})={\varphi}_{\alpha}(x)\ge 0$. As the functional^{} $\widehat{x}$ is continuous^{} in the weak-* topology, we have $\widehat{x}({\varphi}_{\alpha})\to \widehat{x}(\varphi )\ge 0$. Hence $\varphi \in {S}^{+}$. ∎
Property 4.
${\overline{S}}^{+}={S}^{+}$.
Proof.
The inclusion ${\overline{S}}^{+}\subseteq {S}^{+}$ is obvious. And if $\varphi (x)\ge 0$ for all $x\in S$, then by continuity, this holds true for $x\in \overline{S}$ too — so ${\overline{S}}^{+}\supseteq {S}^{+}$. ∎
Properties involving cone inclusion
Property 5 (Farkas’ lemma).
Let $K\mathrm{\subseteq}X$ be a weakly-closed convex cone. Then $x\mathrm{\in}K$ if and only if $\varphi \mathit{}\mathrm{(}x\mathrm{)}\mathrm{\ge}\mathrm{0}$ for all $\varphi \mathrm{\in}{K}^{\mathrm{+}}$.
Proof.
That $\varphi (x)\ge 0$ for $\varphi \in {K}^{+}$ and $x\in K$ is just the definition. For the converse^{}, we show that if $x\in X\setminus K$, then there exists $\varphi \in {K}^{+}$ such that $$.
If $K=\mathrm{\varnothing}$, then the desired $\varphi \in \mathrm{\Phi}={K}^{+}$ exists because $\mathrm{\Phi}$ can separate the points $x$ and $0$. If $K\ne \mathrm{\varnothing}$, by the hyperplane separation theorem, there is a $\varphi \in \mathrm{\Phi}$ such that $$. This $\varphi $ will automatically be in ${K}^{+}$ by Property 2. The zero vector is the weak limit of $ty$, as $t\searrow 0$, for any vector $y$. Thus $0\in K$, and we conclude with ${inf}_{y\in K}\varphi (y)\le 0$. ∎
Property 6.
${K}^{++}=\overline{K}$ for any convex cone $K$. (The anti-cone operation on ${K}^{\mathrm{+}}$ is to be taken with respect to $X$.)
Proof.
We work with $\overline{K}$, which is a weakly-closed convex cone. By Property 5, $x\in \overline{K}$ if and only if $\varphi (x)\ge 0$ for all $\varphi \in {\overline{K}}^{+}={K}^{+}$. But by definition of the second anti-cone, $\widehat{x}\in {({K}^{+})}^{+}$ if and only if $\varphi (x)=\widehat{x}(\varphi )\ge 0$ for all $\varphi \in {K}^{+}$. ∎
Property 7.
Let $K$ and $L$ be convex cones in $X$, with $K$ weakly closed. Then ${K}^{\mathrm{+}}\mathrm{\subseteq}{L}^{\mathrm{+}}$ if and only if $K\mathrm{\supseteq}L$.
Proof.
$${K}^{+}\subseteq {L}^{+}\u27f9K=\overline{K}={K}^{++}\supseteq {L}^{++}=\overline{L}\supseteq L\u27f9{K}^{+}\subseteq {L}^{+}.\mathit{\u220e}$$ |
References
- 1 B. D. Craven and J. J. Kohila. “Generalizations^{} of Farkas’ Theorem^{}.” SIAM Journal on Mathematical Analysis. Vol. 8, No. 6, November 1977.
- 2 David G. Luenberger. Optimization by Vector Space Methods. John Wiley & Sons, 1969.
Title | anti-cone |
---|---|
Canonical name | Anticone |
Date of creation | 2013-03-22 17:20:48 |
Last modified on | 2013-03-22 17:20:48 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 8 |
Author | stevecheng (10074) |
Entry type | Definition |
Classification | msc 46A03 |
Classification | msc 46A20 |
Synonym | anticone |
Synonym | dual cone |
Related topic | GeneralizedFarkasLemma |