Epi-convergence

In mathematical analysis, epi-convergence is a type of convergence for real-valued and extended real-valued functions.

Epi-convergence is important because it is the appropriate notion of convergence with which to approximate minimization problems in the field of mathematical optimization. The symmetric notion of hypo-convergence is appropriate for maximization problems. Mosco convergence is a generalization of epi-convergence to infinite dimensional spaces.

Definition
Let $$ X $$ be a metric space, and $$ f_{n}: X \to \mathbb{R} $$ a real-valued function for each natural number $$ n $$. We say that the sequence $$ (f^{n}) $$ epi-converges to a function $$ f: X \to \mathbb{R} $$ if for each $$ x \in X $$



\begin{align} & \liminf_{n \to \infty} f_{n}(x_n) \geq f(x) \text{ for every } x_n \to x \text{ and } \\ & \limsup_{n \to \infty} f_n(x_n) \leq f(x) \text{ for some } x_n \to x. \end{align} $$

Extended real-valued extension
The following extension allows epi-convergence to be applied to a sequence of functions with non-constant domain.

Denote by $$ \overline{\mathbb{R}}= \mathbb{R} \cup \{ \pm \infty \} $$ the extended real numbers. Let $$ f_n $$ be a function $$ f_n:X \to \overline{\mathbb{R}} $$ for each $$ n \in \mathbb{N} $$. The sequence $$ (f_n) $$ epi-converges to $$ f: X \to \overline{\mathbb{R}} $$ if for each $$ x \in X $$



\begin{align} & \liminf_{n \to \infty} f_{n}(x_n) \geq f(x) \text{ for every } x_n \to x \text{ and } \\ & \limsup_{n \to \infty} f_n(x_n) \leq f(x) \text{ for some } x_n \to x. \end{align} $$

In fact, epi-convergence coincides with the $\Gamma$-convergence in first countable spaces.

Hypo-convergence
Epi-convergence is the appropriate topology with which to approximate minimization problems. For maximization problems one uses the symmetric notion of hypo-convergence. $$ (f_n) $$ hypo-converges to $$ f $$ if


 * $$\limsup_{n \to \infty} f_n(x_n) \leq f(x) \text{ for every } x_n \to x $$

and


 * $$\liminf_{n \to \infty} f_n(x_n) \geq f(x) \text{ for some } x_n \to x. $$

Relationship to minimization problems
Assume we have a difficult minimization problem


 * $$ \inf_{x \in C} g(x) $$

where $$ g: X \to \mathbb{R} $$ and $$ C \subseteq X $$. We can attempt to approximate this problem by a sequence of easier problems


 * $$ \inf_{x \in C_{n}} g_n(x) $$

for functions $$ g_n $$ and sets $$ C_n $$.

Epi-convergence provides an answer to the question: In what sense should the approximations converge to the original problem in order to guarantee that approximate solutions converge to a solution of the original?

We can embed these optimization problems into the epi-convergence framework by defining extended real-valued functions



\begin{align} f(x) & = \begin{cases} g(x), & x \in C, \\ \infty, & x \not \in C, \end{cases} \\[4pt] f_n(x) & = \begin{cases} g_n(x), & x \in C_n, \\ \infty, & x \not \in C_n. \end{cases} \end{align} $$

So that the problems $$ \inf_{x \in X} f(x) $$ and $$ \inf_{x \in X} f_n(x) $$ are equivalent to the original and approximate problems, respectively.

If $$ (f_n) $$ epi-converges to $$ f $$, then $$ \limsup_{n \to \infty} [\inf f_n] \leq \inf f $$. Furthermore, if $$ x $$ is a limit point of minimizers of $$ f_n $$, then $$ x $$ is a minimizer of $$ f $$. In this sense,


 * $$ \lim_{n \to \infty} \operatorname{argmin} f_n \subseteq \operatorname{argmin} f. $$

Epi-convergence is the weakest notion of convergence for which this result holds.

Properties

 * $$ (f_n) $$ epi-converges to $$ f $$ if and only if $$ (-f_n) $$ hypo-converges to $$ -f $$.
 * $$ (f_n) $$ epi-converges to $$ f $$ if and only if $$ (\operatorname{epi} f_n) $$ converges to $$ \operatorname{epi} f $$ as sets, in the Painlevé–Kuratowski sense of set convergence. Here, $$ \operatorname{epi} f $$ is the epigraph of the function $$ f $$.
 * If $$ f_n $$ epi-converges to $$ f $$, then $$ f $$ is lower semi-continuous.
 * If $$ f_n $$ is convex for each $$ n \in \mathbb{N} $$ and $$ (f_n) $$ epi-converges to $$ f $$, then $$ f $$ is convex.
 * If $$ f^1_{n} \leq f_n \leq f^2_{n} $$ and both $$ (f^1_n) $$ and $$ (f^2_n)$$ epi-converge to  $$ f $$, then  $$ (f_n) $$ epi-converges to  $$ f $$.
 * If $$ (f_n) $$  converges uniformly to  $$ f $$ on each compact set of $$ \mathbb{R}_n $$ and $$ (f_n) $$ are continuous, then $$ (f_n) $$ epi-converges and hypo-converges to  $$ f $$.
 * In general, epi-convergence neither implies nor is implied by pointwise convergence. Additional assumptions can be placed on an pointwise convergent family of functions to guarantee epi-convergence.