In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality.
Definition
[edit]Let be a real topological vector space and let be the dual space to . Denote by
the canonical dual pairing, which is defined by
For a function taking values on the extended real number line, its convex conjugate is the function
whose value at is defined to be the supremum:
or, equivalently, in terms of the infimum:
This definition can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes.[1]
Examples
[edit]For more examples, see § Table of selected convex conjugates.
- The convex conjugate of an affine function is
- The convex conjugate of a power function is
- The convex conjugate of the absolute value function is
- The convex conjugate of the exponential function is
The convex conjugate and Legendre transform of the exponential function agree except that the domain of the convex conjugate is strictly larger as the Legendre transform is only defined for positive real numbers.
Connection with expected shortfall (average value at risk)
[edit]Let F denote a cumulative distribution function of a random variable X. Then (integrating by parts), has the convex conjugate
Ordering
[edit]A particular interpretation has the transform as this is a nondecreasing rearrangement of the initial function f; in particular, for f nondecreasing.
Properties
[edit]The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.
Order reversing
[edit]Declare that if and only if for all Then convex-conjugation is order-reversing, which by definition means that if then
For a family of functions it follows from the fact that supremums may be interchanged that
and from the max–min inequality that
Biconjugate
[edit]The convex conjugate of a function is always lower semi-continuous. The biconjugate (the convex conjugate of the convex conjugate) is also the closed convex hull, i.e. the largest lower semi-continuous convex function with For proper functions
- if and only if is convex and lower semi-continuous, by the Fenchel–Moreau theorem.
Fenchel's inequality
[edit]For any function f and its convex conjugate f *, Fenchel's inequality (also known as the Fenchel–Young inequality) holds for every and :
Furthermore, the equality holds only when . The proof follows from the definition of convex conjugate:
Convexity
[edit]For two functions and and a number the convexity relation
holds. The operation is a convex mapping itself.
Infimal convolution
[edit]The infimal convolution (or epi-sum) of two functions and is defined as
Let be proper, convex and lower semicontinuous functions on Then the infimal convolution is convex and lower semicontinuous (but not necessarily proper),[2] and satisfies
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.[3]
Maximizing argument
[edit]If the function is differentiable, then its derivative is the maximizing argument in the computation of the convex conjugate:
- and
hence
and moreover
Scaling properties
[edit]If for some , then
Behavior under linear transformations
[edit]Let be a bounded linear operator. For any convex function on
where
is the preimage of with respect to and is the adjoint operator of [4]
A closed convex function is symmetric with respect to a given set of orthogonal linear transformations,
- for all and all
if and only if its convex conjugate is symmetric with respect to
Table of selected convex conjugates
[edit]The following table provides Legendre transforms for many common functions as well as a few useful properties.[5]
(where ) | |||
(where ) | |||
(where ) | (where ) | ||
(where ) | (where ) | ||
See also
[edit]References
[edit]- ^ "Legendre Transform". Retrieved April 14, 2019.
- ^ Phelps, Robert (1993). Convex Functions, Monotone Operators and Differentiability (2 ed.). Springer. p. 42. ISBN 0-387-56715-1.
- ^ Bauschke, Heinz H.; Goebel, Rafal; Lucet, Yves; Wang, Xianfu (2008). "The Proximal Average: Basic Theory". SIAM Journal on Optimization. 19 (2): 766. CiteSeerX 10.1.1.546.4270. doi:10.1137/070687542.
- ^ Ioffe, A.D. and Tichomirov, V.M. (1979), Theorie der Extremalaufgaben. Deutscher Verlag der Wissenschaften. Satz 3.4.3
- ^ Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. pp. 50–51. ISBN 978-0-387-29570-1.
- Arnol'd, Vladimir Igorevich (1989). Mathematical Methods of Classical Mechanics (Second ed.). Springer. ISBN 0-387-96890-3. MR 0997295.
- Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.
- Rockafellar, R. Tyrell (1970). Convex Analysis. Princeton: Princeton University Press. ISBN 0-691-01586-4. MR 0274683.
Further reading
[edit]- Touchette, Hugo (2014-10-16). "Legendre-Fenchel transforms in a nutshell" (PDF). Archived from the original (PDF) on 2017-04-07. Retrieved 2017-01-09.
- Touchette, Hugo (2006-11-21). "Elements of convex analysis" (PDF). Archived from the original (PDF) on 2015-05-26. Retrieved 2008-03-26.
- "Legendre and Legendre-Fenchel transforms in a step-by-step explanation". Retrieved 2013-05-18.
- Ellerman, David Patterson (1995-03-21). "Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics". Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics (PDF). The worldly philosophy: studies in intersection of philosophy and economics. Rowman & Littlefield Publishers, Inc. pp. 237–268. ISBN 0-8476-7932-2. Archived (PDF) from the original on 2016-03-05. Retrieved 2019-08-09.
Series G - Reference, Information and Interdisciplinary Subjects Series
[1] (271 pages)
- Ellerman, David Patterson (May 2004) [1995-03-21]. "Introduction to Series-Parallel Duality" (PDF). University of California at Riverside. CiteSeerX 10.1.1.90.3666. Archived from the original on 2019-08-10. Retrieved 2019-08-09. [2] (24 pages)