Vfunktion

Vfunktion is a term used in mathematics and related disciplines to denote a class of simple, V-shaped functions that are often used as loss or penalty terms in optimization and statistics. In its basic form, a Vfunktion is a piecewise-linear, convex function of a real variable with a single vertex. The canonical member is f(x) = α|x − β| + γ, where α ≥ 0 controls the slope, β is the location of the vertex, and γ shifts the function vertically.

For α > 0 the function is continuous and convex; it is differentiable for x ≠ β, with left-hand slope −α and right-hand slope α. At x = β, the derivative is not defined; instead the subdifferential equals [−α, α]. The function attains its minimum at x = β with value γ. By allowing translations and scale (β and α), the family can model penalties that are symmetric around a target value and grow linearly with deviation.

A generalized form introduces a skew term or different left and right slopes, f(x) = a|x − β| + d, or more generally f(x) = max{m1(x − β) + γ1, m2(x − β) + γ2}, producing a V-shaped curve with adjustable asymmetry. Beyond one dimension, multivariate extensions consider sums of Vfunktion terms for residual components, yielding convex, piecewise-linear penalties common in robust optimization.

In applications, Vfunktion-like penalties are used in robust regression, regularization, and convex optimization as alternatives to quadratic losses. They promote sparsity and outlier resistance similar to L1 penalties while offering intuitive, interpretable geometry. They also appear in machine learning models as simple loss terms and in economic models describing costs that rise linearly with deviation from a target.

See also: Absolute value function, convex analysis, piecewise-linear function, proximal operator, L1 norm.