Proper convex function

Definitions

Suppose that f : X \to [-\infty, \infty] is a function taking values in the extended real number line [-\infty, \infty] = \mathbb{R} \cup { \pm\infty }. If f is a convex function or if a minimum point of f is being sought, then f is called proper if

:f(x) -\infty for every x \in X

and if there also exists some point x_0 \in X such that

:f\left( x_0 \right)

That is, a function is proper if it never attains the value -\infty and its effective domain is nonempty. This means that there exists some x \in X at which f(x) \in \mathbb{R} and f is also never equal to -\infty. Convex functions that are not proper are called improper convex functions.

A proper concave function is by definition, any function g : X \to [-\infty, \infty] such that f := -g is a proper convex function. Explicitly, if g : X \to [-\infty, \infty] is a concave function or if a maximum point of g is being sought, then g is called proper if its domain is not empty, it never takes on the value +\infty, and it is not identically equal to -\infty.

Properties

For every proper convex function f : \mathbb{R}^n \to [-\infty, \infty], there exist some b \in \mathbb{R}^n and r \in \mathbb{R} such that

:f(x) \geq x \cdot b - r

for every x \in \mathbb{R}^n.

The sum of two proper convex functions is convex, but not necessarily proper. For instance if the sets A \subset X and B \subset X are non-empty convex sets in the vector space X, then the characteristic functions I_A and I_B are proper convex functions, but if A \cap B = \varnothing then I_A + I_B is identically equal to +\infty.

The infimal convolution of two proper convex functions is convex but not necessarily proper convex.

Citations

References

References

1. (2007). "Infinite Dimensional Analysis: A Hitchhiker's Guide". Springer.
2. Rockafellar, R. Tyrrell. (1997). "Convex Analysis". Princeton University Press.
3. Boyd, Stephen. (2004). "Convex Optimization". Cambridge University Press.
4. (2009). "Theory of extremal problems". North-Holland.