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 $f_: X \to \mathbb$ a real-valued function for each natural number $n$. We say that the sequence $(f^)$ epi-converges to a function $f: X \to \mathbb$ if for each $x \in X$

: $\begin & \liminf_ f_(x_n) \geq f(x) \text x_n \to x \text \\ & \limsup_ f_n(x_n) \leq f(x) \text x_n \to x. \end$

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 \cup \$ the extended real numbers. Let $f_n$ be a function $f_n:X \to \overline$ for each $n \in \mathbb$. The sequence $(f_n)$ epi-converges to $f: X \to \overline$ if for each $x \in X$

: $\begin & \liminf_ f_(x_n) \geq f(x) \text x_n \to x \text \\ & \limsup_ f_n(x_n) \leq f(x) \text x_n \to x. \end$

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

: $\limsup_ f_n(x_n) \leq f(x) \text x_n \to x$

and

: $\liminf_ f_n(x_n) \geq f(x) \text x_n \to x.$

Relationship to minimization problems

Assume we have a difficult minimization problem

: $\inf_ g(x)$

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

: $\inf_ g_n(x)$

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
f(x) & = \begin g(x), & x \in C, \\ \infty, & x \not \in C, \end \\ ptf_n(x) & = \begin g_n(x), & x \in C_n, \\ \infty, & x \not \in C_n. \end
\end

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

If $(f_n)$ epi-converges to $f$, then \limsup_ 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,

: $\lim_ \operatorname f_n \subseteq \operatorname f.$

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 $(\operatorname f_n)$ converges to $\operatorname f$ as sets, in the Painlevé–Kuratowski sense of set convergence. Here, $\operatorname f$ 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 $n \in \mathbb$ and $(f_n)$ epi-converges to $f$, then $f$ is convex.
* If $f^1_ \leq f_n \leq f^2_$ 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 $\mathbb_n$ 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.

References

*
*
* {{cite journal
, last1=Attouch , first1=Hedy
, last2=Wets , first2=Roger , authorlink2=Roger Wets
, title=Epigraphical analysis
, journal=Annales de l'Institut Henri Poincaré C
, volume=6
, pages=73–100
, date=1989
, doi=10.1016/S0294-1449(17)30036-7, bibcode=1989AIHPC...6...73A

Mathematical series
Topology of function spaces
Convergence (mathematics)