Recession cone

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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

Relation to asymptotic cone[edit]

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[edit]

See also[edit]

References[edit]

  1. ^ a b c 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. ^ a b c d e Zălinescu, Constantin (2002). Convex analysis in general vector spaces (J). River Edge, NJ: World Scientific Publishing Co., Inc. pp. 6–7. ISBN 981-238-067-1. MR 1921556. 
  4. ^ a b c Kim C. Border. "Sums of sets, etc." (pdf). Retrieved March 7, 2012. 
  5. ^ a b 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.