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New Formulas For The Legendre-Fenchel Transform: J.-B. Hiriart-Urruty, Juan Enrique Martínez-Legaz
We present new formulas for the LEGENDRE-FENCHEL transform of functions. They concern the following three operations: inverting a strictly monotone convex function, post-composing an arbitrary function with a strictly monotone concave function, multiplying two positively valued strictly monotone convex functions.
The new function f* is automatically convex on X*. In Convex Analysis, the transformation f ? f* plays a role similar to that of FOURIER's or LAPLACE's transform in other areas of Analysis. In particular, one cannot get away from it in analyzing the so-called dual versions of a given optimization problem. That explains why the LEGENDRE-FENCHEL transform occupies a key-place in any book on Convex Analysis.
If one performs some operation on given functions f1 and f2, it is natural to ask how is the LEGENDRE-FENCHEL transform of the resulting function calculated from the separate transforms f*1 and f*2. A whole body of calculus rules has therefore been developed from the beginning of the modern era of Convex Analysis, including basic operations such as adding or subtracting f1 and f2, taking the maximum of f1 and f2, performing the infimal convolution (or epigraphical addition) of f1 and f2, etc. Although less important than these key ones, the operations we consider in the present paper arise incidentally in dealing with some Convex Analysis or optimization problems. In the second section, we calculate the LEGENDRE-FENCHEL transform of the inverse f?1 of a strictly monotone convex function f. The resulting formula relates (f?1)* to f*in a simple way. The third section is devoted to determining the transform of an arbitrary function j post-composed with a strictly monotone concave function g. The obtained expression of (g 0 j)* in terms of g* and j* resembles the existing one when both g and j are convex ([3, Chap X, Theorem 2.5.1]). In the fourth section, we consider the unusual situation (in Convex Analysis) of the product of two positively valued strictly monotone convex functions of the real variable.