Recession cone

In mathematics, especially convex analysis, the recession cone of a set ${\displaystyle A}$ is a cone containing all vectors such that ${\displaystyle A}$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 ${\displaystyle A\subset X}$ for some vector space ${\displaystyle X}$, then the recession cone ${\displaystyle \operatorname {recc} (A)}$ is given by

${\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A,\forall \lambda \geq 0:x+\lambda y\in A\}.}$ ^[2]

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

${\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A:x+y\in A\}.}$ ^[3]

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

${\displaystyle \operatorname {recc} (A)=\bigcap _{t>0}t(A-a)}$ for any choice of ${\displaystyle a\in A.}$ ^[3]

Properties

• If ${\displaystyle A}$ is a nonempty set then ${\displaystyle 0\in \operatorname {recc} (A)}$.
• If ${\displaystyle A}$ is a nonempty convex set then ${\displaystyle \operatorname {recc} (A)}$ is a convex cone. ^[3]
• If ${\displaystyle A}$ is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. ${\displaystyle \mathbb {R} ^{d}}$), then ${\displaystyle \operatorname {recc} (A)=\{0\}}$ if and only if ${\displaystyle A}$ is bounded. ^{[1] [3]}
• If ${\displaystyle A}$ is a nonempty set then ${\displaystyle A+\operatorname {recc} (A)=A}$ where the sum denotes Minkowski addition.

Relation to asymptotic cone

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

${\displaystyle C_{\infty }=\{x\in X:\exists (t_{i})_{i\in I}\subset (0,\infty ),\exists (x_{i})_{i\in I}\subset C:t_{i}\to 0,t_{i}x_{i}\to x\}.}$ ^{[4] [5]}

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

In a finite-dimensional space, then it can be shown that ${\displaystyle C_{\infty }=\operatorname {recc} (C)}$ if ${\displaystyle C}$ 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 ${\displaystyle A,B\subset X}$ a locally convex space, if either ${\displaystyle A}$ or ${\displaystyle B}$ is locally compact and ${\displaystyle \operatorname {recc} (A)\cap \operatorname {recc} (B)}$ is a linear subspace, then ${\displaystyle A-B}$ is closed. ^{[7] [3]}
• Let nonempty closed convex sets ${\displaystyle A,B\subset \mathbb {R} ^{d}}$ such that for any ${\displaystyle y\in \operatorname {recc} (A)\backslash \{0\}}$ then ${\displaystyle -y\not \in \operatorname {recc} (B)}$, then ${\displaystyle A+B}$ 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. 77 (1). Springer Netherlands: 209–220. doi:10.1007/bf00940787. ISSN   0022-3239. S2CID   122403313.
7. J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163: 1–3. doi:10.1007/BF02052480. S2CID   119742919.