Convex conjugate

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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][2]

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 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]

See this article for example.

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 ƒ 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 :

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),[3] 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.[4]

Maximizing argument[edit]

If the function is differentiable, then its derivative is the maximizing argument in the computation of the convex conjugate:

and

whence

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 [5]

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.[6]

(where )
(where )
(where ) (where )
(where ) (where )

See also[edit]

References[edit]

  1. ^ "Legendre Transform". Retrieved April 14, 2019.
  2. ^ Nielsen, Frank. "Legendre transformation and information geometry" (PDF).
  3. ^ Phelps, Robert (1991). Convex Functions, Monotone Operators and Differentiability (2 ed.). Springer. p. 42. ISBN 0-387-56715-1.
  4. ^ 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.
  5. ^ Ioffe, A.D. and Tichomirov, V.M. (1979), Theorie der Extremalaufgaben. Deutscher Verlag der Wissenschaften. Satz 3.4.3
  6. ^ 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.

Further reading[edit]