共轭函数
来自维基百科：请尊重原创。本处仅是转载。具体见：http://en.wikipedia.org/wiki/Convex_conjugate
Definition
Let X be a real normed vector space, and let X * be the dual space to X. Denote the dual pairing by
For a functional
taking values on the extended real number line the convex conjugate
is defined in terms of the supremum by
or, equivalently, in terms of the infimum by
This definition can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes. [1]
[edit]Examples
The convex conjugate of an affine function
is
The convex conjugate of a power function
is
where
The convex conjugate of the absolute value function
is
The convex conjugate of the exponential function is
Convex conjugate and Legendre transform of the exponential function agree except that thedomain of the convex conjugate is strictly larger as the Legendre transform is only defined for positive real numbers.
[edit]Connection with average value at risk
Let F denote a cumulative distribution function of a random variable X. Then
has the convex conjugate
[edit]Ordering
A particular interpretation has the transform
as this is a nondecreasing rearrangement of the initial function f; in particular,finc = f for ƒ nondecreasing.
[edit]Properties
The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of apolyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.
Convex-conjugation is order-reversing: if then . Here
[edit]Biconjugate
The convex conjugate of a function is always lower semi-continuous. The biconjugate f * * (the convex conjugate of the convex conjugate) is also theclosed convex hull, i.e. the largest lower semi-continuous convex function with . For proper functions f, f = f** if and only if f is convex and lower semi-continuous.
[edit]Fenchel's inequality
For any function f and its convex conjugate f* Fenchel's inequality (also known as the Fenchel-Young inequality) holds for everyand :
[edit]Maximizing argument
It is interesting to observe that the derivative of the function is the maximizing argument to compute the convex conjugate:
- and
whence
and moreover
[edit]Scaling properties
If, for some β > 0, , then
In case of an additional parameter (α, say) moreover
where is chosen to be the maximizing argument.
[edit]Behavior under linear transformations
Let A be a linear transformation from Rn to Rm. For any convex function f on Rn, one has
where A* is the adjoint operator of A defined by
A closed convex function f is symmetric with respect to a given set G of orthogonal linear transformations,
if and only if its convex conjugate f* is symmetric with respect toG.
[edit]Infimal convolution
The infimal convolution of two functions f and g is defined as
Let f1, …, fm be proper convex functions onRn. Then
The infimal convolution of two functions has a geometric interpretation: The (strict)epigraph of the infimal convolution of two functions is the Minkowski sum of the (strict) epigraphs of those functions.[1]
[edit]See also
- Dual problem
- Fenchel's duality theorem
- Legendre transformation
- Young's inequality
[edit]References
- ^Bauschke, Heinz H.; Goebel, Rafal; Lucet, Yves; Wang, Xianfu (2008). "The Proximal Average: Basic Theory".SIAM Journal on Optimization 19 (2): 766. doi:10.1137/070687542.
- Arnol'd, Vladimir Igorevich (1989).Mathematical Methods of Classical Mechanics (Second ed.). Springer. ISBN 0-387-96890-3.MR997295.
- Rockafellar, R. Tyrell (1970).Convex Analysis. Princeton: Princeton University Press. ISBN 0-691-01586-4.MR0274683.
[edit]External links
- Touchette, Hugo (2005-07-27). "Legendre-Fenchel transforms in a nutshell" (PDF). http://www.maths.qmw.ac.uk/~ht/archive/lfth2.pdf. Retrieved 2007-07-24. [dead link]
- Touchette, Hugo (2006-11-21). "Elements of convex analysis" (PDF). http://www.maths.qmul.ac.uk/~ht/archive/convex1.pdf. Retrieved 2008-03-26.