In mathematics, especially convex analysis, the recession cone of a set is a cone containing all vectors such that 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 for some vector space , then the recession cone is given by
If is additionally a convex set then the recession cone can equivalently be defined by
If is a nonempty closed convex set then the recession cone can equivalently be defined as
- for any choice of [3]
Properties
- If is a nonempty set then .
- If is a nonempty convex set then is a convex cone.[3]
- If is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. ), then if and only if is bounded.[1][3]
- If is a nonempty set then where the sum denotes Minkowski addition.
Relation to asymptotic cone
The asymptotic cone for is defined by
By the definition it can easily be shown that [4]
In a finite-dimensional space, then it can be shown that if 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 locally convex space, if either or is locally compact and is a linear subspace, then is closed.[7][3]
- Let nonempty closed convex sets such that for any then , then is closed.[1][4]
See also
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. S2CID 122403313.
- ↑ J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163: 1–3. doi:10.1007/BF02052480. S2CID 119742919.