Convex Function

A Convex Function is a real-valued function over an interval such that the line segment between any two points on the graph of the function lies above the graph, in an Euclidean space (or more generally a vector space) of at least two dimensions.

• Context:
  • It can range from being a Weakly-Convex Function to being a Strongly-Convex Function.
  • It can be an input to a Convex Optimization Task.
  • …
• Example(s):
  • a Polynomial Function, [math]\displaystyle{ x^\alpha }[/math] for [math]\displaystyle{ x \gt 0 }[/math] and α ≥ 1 or α ≤ 0;
  • [math]\displaystyle{ \mid x \mid^α }[/math], for α ≥ 1
  • an Exponential Function, [math]\displaystyle{ f(x)=e^x }[/math] for any real number x.
  • [math]\displaystyle{ −\log x }[/math].
  • [math]\displaystyle{ x \log x }[/math].
  • Norm Functions.
  • quadratic-over-linear function [math]\displaystyle{ x^T x/t }[/math] for t > 0
  • a Convex Loss Function.
  • …
• Counter-Example(s):
  • a Non-Convex Function.
  • a Concave Function, such as a geometric mean function.
• See: Convex Space, Vector Space, Convex Set, Quadratic Function, Exponential Function, Jensen's Inequality.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/convex_function Retrieved:2015-1-19.
  • In mathematics, a real-valued function defined on an interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above the graph, in an Euclidean space (or more generally a vector space) of at least two dimensions. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. Well-known examples of convex functions are the quadratic function [math]\displaystyle{ f(x)=x^2 }[/math] and the exponential function [math]\displaystyle{ f(x)=e^x }[/math] for any real number x.

    Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a (strictly) convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and, as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always less than or equal to the expected value of the convex function of the random variable. This result, known as Jensen's inequality, underlies many important inequalities (including, for instance, the arithmetic–geometric mean inequality and Hölder's inequality).

    Exponential growth is a special case of convexity. Exponential growth narrowly means "increasing at a rate proportional to the current value", while convex growth generally means "increasing at an increasing rate (but not necessarily proportionally to current value)".

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