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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 or Fenchel transformation (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality.


Let   be a real topological vector space, and let   be the dual space to  . Denote the dual pairing by


For a function


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][2]


The convex conjugate of an affine function




The convex conjugate of a power function





The convex conjugate of the absolute value function




The convex conjugate of the exponential function   is


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



A particular interpretation has the transform


as this is a nondecreasing rearrangement of the initial function f; in particular,   for ƒ nondecreasing.


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 reversingEdit

Convex-conjugation is order-reversing: if   then  . Here


For a family of functions   it follows from the fact that supremums may be interchanged that


and from the max–min inequality that



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 f,

  if and only if f is convex and lower semi-continuous by Fenchel–Moreau theorem.

Fenchel's inequalityEdit

For any function f and its convex conjugate f *, Fenchel's inequality (also known as the Fenchel–Young inequality) holds for every xX and pX * :


The proof follows immediately from the definition of convex conjugate:  .


For two functions   and   and a number   the convexity relation


holds. The   operation is a convex mapping itself.

Infimal convolutionEdit

The infimal convolution (or epi-sum) of two functions f and g is defined as


Let f1, …, fm be proper, convex and lower semicontinuous functions on Rn. 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 argumentEdit

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




and moreover


Scaling propertiesEdit

If, for some  ,  , then


In case of an additional parameter (α, say) moreover[clarification needed]


where   is chosen to be the maximizing argument.

Behavior under linear transformationsEdit

Let A be a bounded linear operator from X to Y. For any convex function f on X, one has




is the preimage of f w.r.t. A and A* is the adjoint operator of A.[5]

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 to G.

Table of selected convex conjugatesEdit

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 alsoEdit


  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 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.

External linksEdit