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]

See also

• Barrier cone

References

1. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6.
2. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 978-0-387-29570-1.
3. 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.
4. Kim C. Border. "Sums of sets, etc." (pdf). Retrieved March 7, 2012.
5. Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9.
6. 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.
7. J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann. 163.