Dual cone and polar cone

Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.

A set ${\displaystyle C}$ and its dual cone ${\displaystyle C^{*}}$.

A set ${\displaystyle C}$ and its polar cone ${\displaystyle C^{o}}$. The dual cone and the polar cone are symmetric to each other with respect to the origin.

Dual cone

The dual cone ${\displaystyle C^{*}}$  of a subset ${\displaystyle C}$  in a linear space ${\displaystyle X}$ , e.g. Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ , with topological dual space ${\displaystyle X^{*}}$  is the set

${\displaystyle C^{*}=\left\{y\in X^{*}:\langle y,x\rangle \geq 0\quad \forall x\in C\right\},}$ 

where ${\displaystyle \langle y,x\rangle }$  is the duality pairing between ${\displaystyle X}$  and ${\displaystyle X^{*}}$ , i.e. ${\displaystyle \langle y,x\rangle =y(x)}$ .

${\displaystyle C^{*}}$  is always a convex cone, even if ${\displaystyle C}$  is neither convex nor a cone.

When ${\displaystyle C}$  is a cone, the following properties hold:^[1]

• A non-zero vector ${\displaystyle y}$  is in ${\displaystyle C^{*}}$  if and only if both of the following conditions hold: (i) ${\displaystyle y}$  is a normal at the origin of a hyperplane that supports ${\displaystyle C}$ . (ii) ${\displaystyle y}$  and ${\displaystyle C}$  lie on the same side of that supporting hyperplane.
• ${\displaystyle C^{*}}$  is closed and convex.
• ${\displaystyle C_{1}\subseteq C_{2}}$  implies ${\displaystyle C_{2}^{*}\subseteq C_{1}^{*}}$ .
• If ${\displaystyle C}$  has nonempty interior, then ${\displaystyle C^{*}}$  is pointed, i.e. ${\displaystyle C^{*}}$  contains no line in its entirety.
• If ${\displaystyle C}$  is a cone and the closure of ${\displaystyle C}$  is pointed, then ${\displaystyle C^{*}}$  has nonempty interior.
• ${\displaystyle C^{**}}$  is the closure of the smallest convex cone containing ${\displaystyle C}$ .

Self-dual cones

A cone ${\displaystyle C}$  in a vector space ${\displaystyle X}$  is said to be self-dual if ${\displaystyle X}$  can be equipped with an inner product ${\displaystyle \langle .,.\rangle }$  such that the internal dual cone relative to this inner product,

${\displaystyle C_{internal}^{*}:=\left\{y\in X:\langle y,x\rangle \geq 0\quad \forall x\in C\right\},}$ 

is equal to ${\displaystyle C}$ ^[2]. Many authors define the dual cone in the context of a real Hilbert space, (such as ${\displaystyle \mathbb {R} ^{n}}$  equipped with the Euclidean inner product) to be what we have called the internal dual cone, and say a cone is self-dual if it is equal to its internal dual. (This is slightly different than the above definition, which permits a change of inner product. For instance, the above definition makes a cone in ${\displaystyle \mathbb {R} ^{n}}$  with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a with spherical base in ${\displaystyle \mathbb {R} ^{n}}$  is equal to its internal dual.)

The nonnegative orthant of ${\displaystyle \mathbb {R} ^{n}}$  and the space of all positive semidefinite matrices 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 ${\displaystyle \mathbb {R} ^{3}}$  whose base is the convex hull of a regular polygon with an odd number of vertices. A less regular example is the cone in ${\displaystyle \mathbb {R} ^{3}}$  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.

Polar cone

 

The polar of the closed convex cone ${\displaystyle C}$  is the closed convex cone ${\displaystyle C^{o},}$  and vice-versa.

For a set ${\displaystyle C}$  in ${\displaystyle X}$ , the polar cone of ${\displaystyle C}$  is the set

${\displaystyle C^{o}=\left\{y\in X^{*}:\langle y,x\rangle \leq 0\quad \forall x\in C\right\}.}$ ^[3]

It can be seen that the polar cone cone is equal to the negative of the dual cone, i.e. ${\displaystyle C^{o}=-C^{*}}$ .

For a closed convex cone ${\displaystyle C}$  in ${\displaystyle X}$ , the polar cone is equivalent to the polar set for ${\displaystyle C}$ .^[4]

See also

• Bipolar theorem
• Polar set

References

1. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 51–53. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
2. Iochum, Bruno, "Cônes autopolaires et algèbres de Jordan", Springer, 1984.
3. Rockafellar, R. Tyrrell (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. pp. 121–122. ISBN 978-0-691-01586-6.
4. Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.

• Goh, C. J. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. ISBN 0-415-27479-6. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Boltyanski, V. G. (1997). Excursions into combinatorial geometry. New York: Springer. ISBN 3-540-61341-2. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

• Ramm, A.G. (2000). Operator theory and its applications. Providence, R.I.: American Mathematical Society. ISBN 0-8218-1990-9. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)