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.
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
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:
Convexityedit
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 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 argumentedit
If the function is differentiable, then its derivative is the maximizing argument in the computation of the convex conjugate:
^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. Tyrell (1970). Convex Analysis. Princeton: Princeton University Press. ISBN 0-691-01586-4. MR 0274683.
Further readingedit
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)