Convex analysis

Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.

A convex set is a set C ⊆ X, for some vector space X, such that for any x, y ∈ C and λ ∈ [0, 1] then

A convex function is any extended real-valued function f : X → R ∪ {±∞} which satisfies Jensen's inequality, i.e. for any x, y ∈ X and any λ ∈ [0, 1] then

Equivalently, a convex function is any (extended) real valued function such that its epigraph

is a convex set.

The convex conjugate of an extended real-valued (not necessarily convex) function f : X → R ∪ {±∞} is f* : X* → R ∪ {±∞} where X* is the dual space of X, and

The biconjugate of a function f : X → R ∪ {±∞} is the conjugate of the conjugate, typically written as f** : X → R ∪ {±∞}. The biconjugate is useful for showing when strong or weak duality hold (via the perturbation function).

For any x ∈ X the inequality f**(x) ≤ f(x) follows from the Fenchel–Young inequality. For proper functions, f = f** if and only if f is convex and lower semi-continuous by Fenchel–Moreau theorem.

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