Symbolic Fenchel Conjugation

Of key importance in convex analysis and optimization is the notion of duality, and in particular that of Fenchel duality. This work explores improvements to existing algorithms for the symbolic calculation of subdifferentials and Fenchel conjugates of convex functions defined on the real line. More importantly, these algorithms are extended to enable the symbolic calculation of Fenchel conjugates on a class of real-valued functions defined on R. These algorithms are realized in the form of the Maple package SCAT. 1 Background and Motivation To make the development available to a wide variety of practitioners we include the following brief discussion on the basics of convex analysis. 1.1 Definition and Basic Results Suppose f is a function defined on R that takes on values in (−∞,∞] = R ∪ {∞} = R̄. Recall that f is convex if f(λx1 + (1− λ)x2) ≤ λf(x1) + (1− λ)f(x2), for every x1, x2 ∈ R, and all λ ∈ [0, 1]. Recall also that the effective domain of f , dom f , is the set of all points where f is finite-valued. Convex functions lie at the heart of convex, functional and real analysis, as well convex optimization. Several excellent overviews of the subject are available, ranging from Rockafellar’s [13] and Luenberger’s [12] classics, to more modern treatments by Boyd and Vandenberghe [7] and by Borwein and Lewis [5]. Calculus teaches that a minimizer x̄ of a differentiable function f is necessarily a critical point: ∇f(x̄) = 0. Since many interesting functions are not everywhere differentiable, this leads naturally to a generalized derivative, the subdifferential of f at x by ∂f(x) = {y ∈ R : 〈y, x′ − x〉 ≤ f(x′)− f(x), ∀x′ ∈ Rn}.