recession cone

In mathematics, especially convex analysis, the recession cone of a set $A$ is a cone containing all vectors such that $A$ recedes in that direction. That is, the set extends outward in all the directions given by the recession cone.{{cite book|author=Rockafellar, R. Tyrrell|author-link=Rockafellar, R. Tyrrell|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|orig-year=1970|isbn=978-0-691-01586-6|pages=60–76}}

Mathematical definition

Given a nonempty set $A \subset X$ for some vector space $X$ , then the recession cone $\operatorname{recc}(A)$ is given by

: $\operatorname{recc}(A) = \{y \in X: \forall x \in A, \forall \lambda \geq 0: x + \lambda y \in A\}.$ {{cite book |last1=Borwein |first1=Jonathan |last2=Lewis |first2=Adrian |title=Convex Analysis and Nonlinear Optimization: Theory and Examples| edition=2 |year=2006 |publisher=Springer |isbn=978-0-387-29570-1}}

If $A$ is additionally a convex set then the recession cone can equivalently be defined by

: $\operatorname{recc}(A) = \{y \in X: \forall x \in A: x + y \in A\}.$ {{cite book |last=Zălinescu |first=Constantin |title=Convex analysis in general vector spaces |url=https://archive.org/details/convexanalysisge00zali_858 |url-access=limited |publisher=World Scientific Publishing Co., Inc. |isbn=981-238-067-1 |mr=1921556 |issue=J |year=2002 |location=River Edge, NJ |pages=[https://archive.org/details/convexanalysisge00zali_858/page/n25 6]–7}}

If $A$ is a nonempty closed convex set then the recession cone can equivalently be defined as

: $\operatorname{recc}(A) = \bigcap_{t > 0} t(A - a)$ for any choice of $a \in A.$

Properties

If $A$ is a nonempty set then $0 \in \operatorname{recc}(A)$ .
If $A$ is a nonempty convex set then $\operatorname{recc}(A)$ is a convex cone.
If $A$ is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. $\mathbb{R}^d$ ), then $\operatorname{recc}(A) = \{0\}$ if and only if $A$ is bounded.
If $A$ is a nonempty set then $A + \operatorname{recc}(A) = A$ where the sum denotes Minkowski addition.

Relation to asymptotic cone

The asymptotic cone for $C \subseteq X$ is defined by

: $C_{\infty} = \{x \in X: \exists (t_i)_{i \in I} \subset (0,\infty), \exists (x_i)_{i \in I} \subset C: t_i \to 0, t_i x_i \to x\}.$ {{cite web|url=http://www.hss.caltech.edu/~kcb/Notes/AsymptoticCones.pdf|title=Sums of sets, etc.|author=Kim C. Border|accessdate=March 7, 2012}}{{cite book|title=Asymptotic cones and functions in optimization and variational inequalities|url=https://archive.org/details/asymptoticconesf00ausl|url-access=limited|author=Alfred Auslender|author2=M. Teboulle|publisher=Springer|year=2003|isbn=978-0-387-95520-9|pages=[https://archive.org/details/asymptoticconesf00ausl/page/n36 25]–80}}

By the definition it can easily be shown that $\operatorname{recc}(C) \subseteq C_\infty.$

In a finite-dimensional space, then it can be shown that $C_{\infty} = \operatorname{recc}(C)$ if $C$ is nonempty, closed and convex. In infinite-dimensional spaces, then the relation between asymptotic cones and recession cones is more complicated, with properties for their equivalence summarized in.{{cite journal|title=Recession cones and asymptotically compact sets|last=Zălinescu |first=Constantin|journal=Journal of Optimization Theory and Applications|publisher=Springer Netherlands|issn=0022-3239|pages=209–220|volume=77|issue=1|year=1993|doi=10.1007/bf00940787|s2cid=122403313 }}

Sum of closed sets

Dieudonné's theorem: Let nonempty closed convex sets $A,B \subset X$ a locally convex space, if either $A$ or $B$ is locally compact and $\operatorname{recc}(A) \cap \operatorname{recc}(B)$ is a linear subspace, then $A - B$ is closed.{{cite journal|title=Sur la séparation des ensembles convexes|author=J. Dieudonné|year=1966|journal=Math. Ann.|volume=163|pages=1–3|doi=10.1007/BF02052480|s2cid=119742919}}
Let nonempty closed convex sets $A,B \subset \mathbb{R}^d$ such that for any $y \in \operatorname{recc}(A) \backslash \{0\}$ then $-y \not\in \operatorname{recc}(B)$ , then $A + B$ is closed.

References

Category:Convex analysis