47:
976:
31:
936:. Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual. This is slightly different from the above definition, which permits a change of inner product. For instance, the above definition makes a cone in
484:
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226:
382:
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whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.
320:
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803:
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are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones"). So are all cones in
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1399:
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with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in
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1447:
479:{\displaystyle C^{\prime }:=\left\{f\in X^{\prime }:\operatorname {Re} \left(f(x)\right)\geq 0{\text{ for all }}x\in C\right\}}
1537:
1391:
963:
whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in
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683:{\displaystyle C_{\text{internal}}^{*}:=\left\{y\in X:\langle y,x\rangle \geq 0\quad \forall x\in C\right\}.}
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1099:{\displaystyle C^{o}=\left\{y\in X^{*}:\langle y,x\rangle \leq 0\quad \forall x\in C\right\}.}
221:{\displaystyle C^{*}=\left\{y\in X^{*}:\langle y,x\rangle \geq 0\quad \forall x\in C\right\},}
1588:
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58:. The dual cone and the polar cone are symmetric to each other with respect to the origin.
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30:
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1394:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
1109:
It can be seen that the polar cone is equal to the negative of the dual cone, i.e.
932:⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to
112:
105:
1471:
341:
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equipped with the
Euclidean inner product) to be what is sometimes called the
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1356:
1341:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
1623:
1150:
1129:
491:
1268:
1221:
Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.
581:
Alternatively, many authors define the dual cone in the context of a real
17:
948:
98:
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974:
45:
29:
1420:
1238:. Princeton, NJ: Princeton University Press. pp. 121–122.
558:
545:
514:
415:
391:
1364:
Ramm, A.G. (2000). Shivakumar, P.N.; Strauss, A.V. (eds.).
372:
is the following set of continuous linear functionals on
895:
is the closure of the smallest convex cone containing
1016:
811:
771:
718:
if and only if both of the following conditions hold:
602:
539:
508:
385:
281:
237:
138:
1320:
Duality in optimization and variational inequalities
1679:
1658:
1637:
1571:
1505:
1454:
1368:. Providence, R.I.: American Mathematical Society.
1261:
751:
lie on the same side of that supporting hyperplane.
1098:
847:
797:
682:
565:
521:
478:
314:
255:
220:
1185:Boyd, Stephen P.; Vandenberghe, Lieven (2004).
1337:Narici, Lawrence; Beckenstein, Edward (2011).
1194:. Cambridge University Press. pp. 51–53.
1432:
1172:
8:
1066:
1054:
848:{\displaystyle C_{2}^{*}\subseteq C_{1}^{*}}
650:
638:
294:
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238:
188:
176:
360:over the real or complex numbers, then the
1439:
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1417:
1322:. London; New York: Taylor & Francis.
706:is a cone, the following properties hold:
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315:{\displaystyle \langle y,x\rangle =y(x)}
1259:Aliprantis, C.D.; Border, K.C. (2007).
1162:
577:In a Hilbert space (internal dual cone)
566:{\displaystyle C^{\prime }=X^{\prime }}
1301:Excursions into combinatorial geometry
1128:, the polar cone is equivalent to the
1263:(3 ed.). Springer. p. 215.
7:
1366:Operator theory and its applications
979:The polar of the closed convex cone
798:{\displaystyle C_{1}\subseteq C_{2}}
256:{\displaystyle \langle y,x\rangle }
1299:; Martini, H.; Soltan, P. (1997).
1076:
660:
198:
25:
1448:Ordered topological vector spaces
874:contains no line in its entirety.
698:Using this latter definition for
69:are closely related concepts in
1318:Goh, C. J.; Yang, X.Q. (2002).
1075:
944:is equal to its internal dual.
659:
197:
957:positive semidefinite matrices
443:
437:
309:
303:
1:
1538:Locally convex vector lattice
901:hyperplane separation theorem
881:is a cone and the closure of
348:In a topological vector space
1386:; Wolff, Manfred P. (1999).
862:has nonempty interior, then
529:will be a convex cone. If
522:{\displaystyle C^{\prime }}
27:Concepts in convex analysis
1734:
1492:Topological vector lattice
983:is the closed convex cone
1572:Types of elements/subsets
1388:Topological Vector Spaces
1339:Topological Vector Spaces
1173:Schaefer & Wolff 1999
1120:For a closed convex cone
1487:Positive linear operator
928:can be equipped with an
358:topological vector space
1467:Partially ordered space
1232:Rockafellar, R. Tyrrell
1638:Topologies/Convergence
1506:Types of orders/spaces
1303:. New York: Springer.
1100:
988:
899:(a consequence of the
889:has nonempty interior.
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799:
684:
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59:
43:
1269:10.1007/3-540-29587-9
1101:
978:
955:and the space of all
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1687:Freudenthal spectral
1619:Quasi-interior point
1462:Ordered vector space
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809:
769:
702:, we have that when
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383:
279:
235:
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1384:Schaefer, Helmut H.
1188:Convex Optimization
1175:, pp. 215–222.
844:
826:
730:at the origin of a
617:
459: for all
54:and its polar cone
1718:Linear programming
1096:
989:
916:in a vector space
845:
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795:
710:A non-zero vector
680:
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591:internal dual cone
563:
519:
498:. No matter what
476:
312:
253:
218:
60:
44:
38:and its dual cone
1695:
1694:
1645:Order convergence
1563:Regularly ordered
1401:978-1-4612-7155-0
1297:Boltyanski, V. G.
1278:978-3-540-32696-0
1245:978-0-691-01586-6
1201:978-0-521-83378-3
987:, and vice versa.
885:is pointed, then
610:
460:
86:In a vector space
16:(Redirected from
1725:
1589:Lattice disjoint
1548:Order bound dual
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947:The nonnegative
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1713:Convex geometry
1708:Convex analysis
1698:
1697:
1696:
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1633:
1629:Weak order unit
1594:Dual/Polar cone
1567:
1533:Fréchet lattice
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1236:Convex Analysis
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1146:Bipolar theorem
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908:Self-dual cones
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1650:Order topology
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1609:Order summable
1606:
1604:Order complete
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1584:Cone-saturated
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1558:Order complete
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1543:Normed lattice
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1528:Banach lattice
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1497:Vector lattice
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1477:Order topology
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1455:Basic concepts
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920:is said to be
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584:
583:Hilbert space
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490:which is the
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1680:Main results
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1290:Bibliography
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1205:. Retrieved
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494:of the set -
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328:is always a
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106:linear space
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39:
35:
1599:Normal cone
1523:Archimedean
1472:Riesz space
1207:October 15,
1007:is the set
763:and convex.
533:⊆ {0} then
336:is neither
330:convex cone
129:is the set
75:mathematics
1702:Categories
1614:Order unit
1553:Order dual
1482:Order unit
1157:References
1001:polar cone
991:For a set
971:Polar cone
732:hyperplane
694:Properties
332:, even if
124:dual space
67:polar cone
1659:Operators
1624:Solid set
1410:840278135
1357:144216834
1234:(1997) .
1151:Polar set
1130:polar set
1083:∈
1077:∀
1070:≤
1067:⟩
1055:⟨
1047:∗
1039:∈
922:self-dual
841:∗
828:⊆
823:∗
783:⊆
667:∈
661:∀
654:≥
651:⟩
639:⟨
630:∈
614:∗
585:(such as
559:′
546:′
515:′
466:∈
452:≥
427:
416:′
408:∈
392:′
362:dual cone
295:⟩
283:⟨
251:⟩
239:⟨
205:∈
199:∀
192:≥
189:⟩
177:⟨
169:∗
161:∈
145:∗
111:over the
92:dual cone
81:Dual cone
63:Dual cone
18:Dual cone
1666:Positive
1518:AM-space
1513:AL-space
1140:See also
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609:internal
267:between
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912:A cone
870:, i.e.
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263:is the
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