Variational Methods in Convex Analysis

We use variational methods to provide a concise development of a number of basic results in convex and functional analysis. This illuminates the parallels between convex analysis and smooth subdifferential theory. 1. The purpose of this note is to give a concise and explicit account of the following folklore: several fundamental theorems in convex analysis such as the sandwich theorem and the Fenchel duality theorem may usefully be proven by variational arguments. Many important results in linear functional analysis can then be easily deduced as special cases. These are entirely parallel to the basic calculus of smooth subdifferential theory. Some of these relationships have already been discussed in [1, 2, 5, 6, 12, 18]. 2. By a ‘variational argument’ we connote a proof with two main components: (a) an argument that an appropriate auxiliary function attains its minimum and (b) a ‘decoupling’ mechanism in a sense we make precise below. It is well known that this methodology lies behind many basic results of smooth subdifferential theory [6, 20]. It is known, but not always made explicit, that this is equally so in convex analysis. Here we record in an organized fashion that this method also lies behind most of the important theorems in convex analysis. In convex analysis the role of (a) is usually played by the following theorem attributed to Fenchel and Rockafellar (among others) for which some preliminaries are needed. Let X be a real locally convex topological vector space. Recall that the domain of an extended valued convex function f on X (denoted dom f) is the set of points with value less than +∞. A subset T of X is absorbing if X = λ>0 λT and a point s is in the core of a set S ⊂ X (denoted by s ∈ core S) provided that S − s is absorbing and s ∈ S. A symmetric, convex, closed and absorbing subset of X is called a barrel. We say X is barrelled if every barrel of X is a neighborhood of zero. All Baire—and hence all complete metrizable—locally convex spaces are barrelled, but not conversely. Recall that x∗ ∈ X∗, the topological dual, is a subgradient of f : X → (−∞, +∞] at x ∈ dom f provided that f(y)− f(x) ≥ 〈x∗, y − x〉. The set of all subgradients of f at x is called the subdifferential of f at x and is denoted ∂f(x). We use the standard convention that ∂f(x) = ∅ for x 6∈ dom f . We use contf to denote the set of all continuity points of f . Date: November 22, 2006.