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An inequality relating a function and its Fenchel conjugate via the dual pairing.
Fenchel-Young inequality: Let be a proper function, and let be its Fenchel conjugate. Then for all and ,
If is convex, equality holds if and only if (equivalently, ), where denotes the subdifferential.
This inequality is the basic mechanism behind weak duality in convex primal-dual pairs: conjugate-based dual objectives arise by repeatedly applying Fenchel-Young.
A Fenchel conjugate of an extended-real-valued function is the function defined by
where denotes the Euclidean pairing.
This operation (also called the Legendre–Fenchel transform) turns into a (possibly extended-real) convex function, and it is the basic object behind Fenchel–Young inequality and convex duality. The definition uses the supremum over all affine functionals .
A subdifferential of a convex function at a point in its effective domain is the set
Any is called a subgradient of at .
Geometrically, encodes a supporting hyperplane to the epigraph of at , with normal determined by (compare hyperplane). If is differentiable at (in the sense of the derivative / differentiable map), then .
A primal-dual pair in convex optimization is a pair of problems built from convex objectives so that the dual objective gives a universal lower bound on the primal optimal value. A common (Fenchel) form starts from convex functions and and a linear map , and defines
where and are Fenchel conjugates and use infimum and supremum. The inequality (weak duality) follows from the Fenchel-Young inequality, and under suitable regularity conditions one has strong duality , often certified by subgradient conditions.