Maksafunktio

Maksafunktio, or the maximum function, is a mathematical operation that returns the largest element among a finite collection of numbers. In the multivariate form, it is the function M: R^n → R defined by M(x1, ..., xn) = max{x1, ..., xn}. It is a commutative and associative operation and is nondecreasing in each argument: increasing any input cannot decrease the output. It is idempotent: max(x, x) = x. For two numbers a and b, max(a, b) can be written as (a + b + |a − b|)/2, a useful identity in analysis.

More generally, for a vector x, M(x) = max_i x_i. The function is continuous and, since it is the pointwise maximum of a finite set of affine functions, it is convex. It is not differentiable at points where the maximum is attained by more than one coordinate; the subdifferential at such points is the convex hull of the corresponding coordinate unit vectors.

Applications and usage: Maksafunktio is central in optimization and decision theory, for example in formulations that take the maximum cost, risk, or utility. In computer science, the max operation is used in algorithms and in neural networks as the max-pooling layer, which reduces spatial dimensions by selecting the maximum value in each region. In statistics, the maximum is an order statistic describing the largest observation in a sample.

Variants and approximations: The softmax function provides a smooth, differentiable approximation to the maximum and is widely used in machine learning when a differentiable surrogate is needed. Generalizations to infinite sets or different indexings extend the idea via the supremum operation in more abstract settings.