Maxfunktionen

Maxfunktionen, in mathematics commonly called the maximum function, refers to an operation that assigns the largest element from a finite set of numbers or from a family of real-valued functions. For numbers a1, …, an, the value is max(a1, …, an). When considering functions fi on a common domain X, the maximum function is defined by F(x) = maxi fi(x) for x in X. If the index set is infinite, the natural notion becomes the supremum supi fi(x), unless the maximum is attained.

Notation commonly used includes max(a, b) for two arguments and maxi ai for a finite index set. The max operation is extended to vectors by taking the maximum over components, maxj xj.

Key properties include commutativity, associativity, and idempotence (max(a, a) = a). It is monotone: if ai ≤ bi for all i, then maxi ai ≤ maxi bi. The max of a collection of convex functions is convex; however, the max function is generally non-differentiable at points where two or more arguments attain the maximum, leading to a subdifferential in convex analysis.

Examples and applications: ReLU, defined as ReLU(x) = max(0, x), is a simple instance of a max function. The maximum of affine functions yields a convex, piecewise-linear function useful in optimization and machine learning. In programming and algorithms, the max operation is a fundamental primitive; in max-plus algebra it serves as an additive operator.

Related concepts include the minimum function, argmax, and differentiable approximations such as softmax.