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 $x\in X$

${\begin{aligned}&\liminf _{n\to \infty }f_{n}(x_{n})\geq f(x){\text{ for every }}x_{n}\to x{\text{ and }}\\&\limsup _{n\to \infty }f_{n}(x_{n})\leq f(x){\text{ for some }}x_{n}\to x.\end{aligned}}$

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 $x\in X$

${\begin{aligned}&\liminf _{n\to \infty }f_{n}(x_{n})\geq f(x){\text{ for every }}x_{n}\to x{\text{ and }}\\&\limsup _{n\to \infty }f_{n}(x_{n})\leq f(x){\text{ for some }}x_{n}\to x.\end{aligned}}$

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

${\begin{aligned}f(x)&={\begin{cases}g(x),&x\in C,\\\infty ,&x\not \in C,\end{cases}}\\[4pt]f_{n}(x)&={\begin{cases}g_{n}(x),&x\in C_{n},\\\infty ,&x\not \in C_{n}.\end{cases}}\end{aligned}}$

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 $f_{n}^{1}\leq f_{n}\leq f_{n}^{2}$ and both $(f_{n}^{1})$ and $(f_{n}^{2})$ 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.

References

• Rockafellar, R. Tyrrell; Wets, Roger (2009). "Epigraphical Limits". Variational Analysis. Vol. 317. Springer Science & Business Media. doi:10.1007/978-3-642-02431-3_7.
• Kall, Peter (1986). "Approximation to optimization problems: an elementary review". Mathematics of Operations Research. 11 (1): 9–18. doi:10.1287/moor.11.1.9.
• Attouch, Hedy; Wets, Roger (1989). "Epigraphical analysis". Annales de l'Institut Henri Poincaré Analyse non linéaire. 6: 73–100. doi:10.1016/S0294-1449(17)30036-7.