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.^[1]

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\}.

^[2]

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\}.

^[3]

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.

^[3]

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.^[3]
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.^[1]^[3]
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\}.

^[4]^[5]

By the definition it can easily be shown that $\operatorname {recc}(C)\subseteq C_{\infty }.$ ^[4]

In a finite-dimensional space, then it can be shown that $C_{{\infty }}=\operatorname {recc}(C)$ if $C$ is nonempty, closed and convex.^[5] In infinite-dimensional spaces, then the relation between asymptotic cones and recession cones is more complicated, with properties for their equivalence summarized in.^[6]

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.^[7]^[3]
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.^[1]^[4]

References

1 2 3 Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6.
↑ Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 978-0-387-29570-1.
1 2 3 4 5 Zălinescu, Constantin (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 6–7. ISBN 981-238-067-1. MR 1921556.
1 2 3 Kim C. Border. "Sums of sets, etc." (pdf). Retrieved March 7, 2012.
1 2 Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9.
↑ Zălinescu, Constantin (1993). "Recession cones and asymptotically compact sets". Journal of Optimization Theory and Applications. Springer Netherlands. 77 (1): 209–220. doi:10.1007/bf00940787. ISSN 0022-3239.
↑ J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163.

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