Convex conjugate

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. 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 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. For more examples, see . 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. 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 A particular interpretation has the transform as this is a nondecreasing rearrangement of the initial function f; in particular, for f 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. 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 The convex conjugate of a function is always lower semi-continuous.