Fencheldualitet

Fencheldualitet, or Fenchel duality, is a central concept in convex analysis that relates a convex optimization problem to a corresponding dual problem through Fenchel conjugation. For a proper lower semicontinuous convex function f on a vector space, its Fenchel conjugate is defined by f*(y) = sup_x (⟨y, x⟩ − f(x)). The Fenchel conjugate summarizes how f twists under linear measurements and is tied to several inequalities, notably the Fenchel–Young inequality f(x) + f*(y) ≥ ⟨y, x⟩, with equality when y ∈ ∂f(x).

In a typical setting, consider a primal problem of the form min_x f(x) + g(Ax), where f and g are convex lsc and A is a linear operator. The Fenchel dual problem is max_y −f*(A^T y) − g*(−y). Under standard regularity conditions (for example, Slater-type conditions), the optimal values of the primal and dual problems coincide (strong duality), and optimal solutions satisfy subgradient optimality conditions: −A^T y* ∈ ∂f(x*) and y* ∈ ∂g(Ax*).

Related concepts include the biconjugate f** and the Fenchel–Moreau theorem, which states that any proper lsc convex f equals its double conjugate f**. This framework underpins many methods in convex optimization, including proximal algorithms and operator-splitting techniques, and provides a flexible tool for deriving dual formulations and certificates of optimality. Fenchel duality thus serves as a unifying lens for understanding convex problems through conjugate functions and their subdifferentials.