Recession cone

<h2 id="mathematical-definition">Mathematical definition</h2>
Given a nonempty set 
 
 
 
 A
 ⊂
 X
 
 
 {\displaystyle A\subset X}
 
 for some <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
, then the recession cone 
 
 
 
 recc
 ⁡
 (
 A
 )
 
 
 {\displaystyle \operatorname {recc} (A)}
 
 is given by

recc
 ⁡
 (
 A
 )
 =
 {
 y
 ∈
 X
 :
 ∀
 x
 ∈
 A
 ,
 ∀
 λ
 ≥
 0
 :
 x
 +
 λ
 y
 ∈
 A
 }
 .
 
 
 {\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A,\forall \lambda \geq 0:x+\lambda y\in A\}.}
 
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is additionally a <a href="/facts/Convex_set/vdAuJRJl">convex set</a> then the recession cone can equivalently be defined by

recc
 ⁡
 (
 A
 )
 =
 {
 y
 ∈
 X
 :
 ∀
 x
 ∈
 A
 :
 x
 +
 y
 ∈
 A
 }
 .
 
 
 {\displaystyle \operatorname {recc} (A)=\{y\in X:\forall x\in A:x+y\in A\}.}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a nonempty <a href="/facts/Closed_set/upYnRTUj">closed</a> convex set then the recession cone can equivalently be defined as

recc
 ⁡
 (
 A
 )
 =
 
 ⋂
 
 t
 >
 0
 
 
 t
 (
 A
 −
 a
 )
 
 
 {\displaystyle \operatorname {recc} (A)=\bigcap _{t>0}t(A-a)}
 
 for any choice of 
 
 
 
 a
 ∈
 A
 .
 
 
 {\displaystyle a\in A.}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
<h2 id="properties">Properties</h2>
<ul><li>If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a nonempty set then 
 
 
 
 0
 ∈
 recc
 ⁡
 (
 A
 )
 
 
 {\displaystyle 0\in \operatorname {recc} (A)}
 
.</li>
<li>If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a nonempty convex set then 
 
 
 
 recc
 ⁡
 (
 A
 )
 
 
 {\displaystyle \operatorname {recc} (A)}
 
 is a <a href="/facts/Convex_cone/h1tARfw4">convex cone</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a nonempty closed convex subset of a finite-dimensional <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> (e.g. 
 
 
 
 
 
 R
 
 
 d
 
 
 
 
 {\displaystyle \mathbb {R} ^{d}}
 
), then 
 
 
 
 recc
 ⁡
 (
 A
 )
 =
 {
 0
 }
 
 
 {\displaystyle \operatorname {recc} (A)=\{0\}}
 
 if and only if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is bounded.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li>
<li>If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a nonempty set then 
 
 
 
 A
 +
 recc
 ⁡
 (
 A
 )
 =
 A
 
 
 {\displaystyle A+\operatorname {recc} (A)=A}
 
 where the sum denotes <a href="/facts/Minkowski_addition/dEQfoJ1A">Minkowski addition</a>.</li></ul>
<h2 id="relation-to-asymptotic-cone">Relation to asymptotic cone</h2>
The <a href="/facts/Asymptotic_cone/aC31JMos">asymptotic cone</a> for 
 
 
 
 C
 ⊆
 X
 
 
 {\displaystyle C\subseteq X}
 
 is defined by

C
 
 ∞
 
 
 =
 {
 x
 ∈
 X
 :
 ∃
 (
 
 t
 
 i
 
 
 
 )
 
 i
 ∈
 I
 
 
 ⊂
 (
 0
 ,
 ∞
 )
 ,
 ∃
 (
 
 x
 
 i
 
 
 
 )
 
 i
 ∈
 I
 
 
 ⊂
 C
 :
 
 t
 
 i
 
 
 →
 0
 ,
 
 t
 
 i
 
 
 
 x
 
 i
 
 
 →
 x
 }
 .
 
 
 {\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\}.}
 
<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
By the definition it can easily be shown that 
 
 
 
 recc
 ⁡
 (
 C
 )
 ⊆
 
 C
 
 ∞
 
 
 .
 
 
 {\displaystyle \operatorname {recc} (C)\subseteq C_{\infty }.}
 
<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
In a finite-dimensional space, then it can be shown that 
 
 
 
 
 C
 
 ∞
 
 
 =
 recc
 ⁡
 (
 C
 )
 
 
 {\displaystyle C_{\infty }=\operatorname {recc} (C)}
 
 if 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is nonempty, closed and convex.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> In infinite-dimensional spaces, then the relation between asymptotic cones and recession cones is more complicated, with properties for their equivalence summarized in.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="sum-of-closed-sets">Sum of closed sets</h2>
<ul><li><a href="/facts/Dieudonn%25C3%25A9%2527s_theorem/xLK3Nq0c">Dieudonné's theorem</a>: Let nonempty closed convex sets 
 
 
 
 A
 ,
 B
 ⊂
 X
 
 
 {\displaystyle A,B\subset X}
 
 a <a href="/facts/Locally_convex_space/AGrPrayC">locally convex space</a>, if either 
 
 
 
 A
 
 
 {\displaystyle A}
 
 or 
 
 
 
 B
 
 
 {\displaystyle B}
 
 is <a href="/facts/Locally_compact/hesalT7q">locally compact</a> and 
 
 
 
 recc
 ⁡
 (
 A
 )
 ∩
 recc
 ⁡
 (
 B
 )
 
 
 {\displaystyle \operatorname {recc} (A)\cap \operatorname {recc} (B)}
 
 is a <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a>, then 
 
 
 
 A
 −
 B
 
 
 {\displaystyle A-B}
 
 is closed.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a></li>
<li>Let nonempty closed convex sets 
 
 
 
 A
 ,
 B
 ⊂
 
 
 R
 
 
 d
 
 
 
 
 {\displaystyle A,B\subset \mathbb {R} ^{d}}
 
 such that for any 
 
 
 
 y
 ∈
 recc
 ⁡
 (
 A
 )
 ∖
 {
 0
 }
 
 
 {\displaystyle y\in \operatorname {recc} (A)\backslash \{0\}}
 
 then 
 
 
 
 −
 y
 ∉
 recc
 ⁡
 (
 B
 )
 
 
 {\displaystyle -y\not \in \operatorname {recc} (B)}
 
, then 
 
 
 
 A
 +
 B
 
 
 {\displaystyle A+B}
 
 is closed.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Barrier_cone/3mYgmGRs">Barrier cone</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6. <a href="978-0-691-01586-6" target="_blank">978-0-691-01586-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 978-0-387-29570-1. <a href="978-0-387-29570-1" target="_blank">978-0-387-29570-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="981-238-067-1" target="_blank">981-238-067-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">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. <a href="981-238-067-1" target="_blank">981-238-067-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="981-238-067-1" target="_blank">981-238-067-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6. <a href="978-0-691-01586-6" target="_blank">978-0-691-01586-6</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">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. <a href="981-238-067-1" target="_blank">981-238-067-1</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kim C. Border. "Sums of sets, etc" (PDF). Retrieved March 7, 2012. <a href="/wiki/Kim_C._Border" target="_blank">/wiki/Kim_C._Border</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9. <a href="978-0-387-95520-9" target="_blank">978-0-387-95520-9</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Kim C. Border. "Sums of sets, etc" (PDF). Retrieved March 7, 2012. <a href="/wiki/Kim_C._Border" target="_blank">/wiki/Kim_C._Border</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Alfred Auslender; M. Teboulle (2003). Asymptotic cones and functions in optimization and variational inequalities. Springer. pp. 25–80. ISBN 978-0-387-95520-9. <a href="978-0-387-95520-9" target="_blank">978-0-387-95520-9</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163: 1–3. doi:10.1007/BF02052480. S2CID 119742919. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">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. <a href="981-238-067-1" target="_blank">981-238-067-1</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 60–76. ISBN 978-0-691-01586-6. <a href="978-0-691-01586-6" target="_blank">978-0-691-01586-6</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Kim C. Border. "Sums of sets, etc" (PDF). Retrieved March 7, 2012. <a href="/wiki/Kim_C._Border" target="_blank">/wiki/Kim_C._Border</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
</ol>

Recession cone open-in-new

Recession cone