Convex analysis

Convex analysis is the branch of mathematics that studies convex sets, convex functions, and their applications to optimization, functional analysis, variational analysis, convex geometry, economics, and related fields. A set is convex if it contains every line segment joining two of its points. A function is convex if its value at a weighted average of two points is no greater than the corresponding weighted average of its values. Informally, convex sets have no inward dents, and convex functions have graphs that bend upward.

Convexity implies certain global features of a problem. For example, in a convex optimization problem, every local minimum is also a global minimum. Convex sets can often be separated by hyperplanes, and convex functions can be studied through supporting affine functions. Convex analysis is a common thread in modern optimization, duality theory, and the study of nonsmooth problems. The tools of convex analysis include the epigraph of a function, the subdifferential, the Legendre–Fenchel transform, and the Fenchel–Moreau theorem. These allow many constrained problems to be rewritten in geometric form and many optimization problems to be paired with dual problems.