Recession cone

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}.

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}.

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.

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.
• 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.
• 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}.

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

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

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.
• 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.

References

References

1. Rockafellar, R. Tyrrell. (1997). "Convex Analysis". Princeton University Press.
2. (2006). "Convex Analysis and Nonlinear Optimization: Theory and Examples". Springer.
3. Zălinescu, Constantin. (2002). "Convex analysis in general vector spaces". World Scientific Publishing Co., Inc..
4. [[Kim C. Border]]. "Sums of sets, etc.".
5. Alfred Auslender. (2003). "Asymptotic cones and functions in optimization and variational inequalities". Springer.
6. Zălinescu, Constantin. (1993). "Recession cones and asymptotically compact sets". Springer Netherlands.
7. J. Dieudonné. (1966). "Sur la séparation des ensembles convexes". Math. Ann..