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

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

Extended real-valued extension

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

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

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

In fact, epi-convergence coincides with the ${\displaystyle \Gamma }$-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. ${\displaystyle (f_{n})}$ hypo-converges to ${\displaystyle f}$ if

${\displaystyle \limsup _{n\to \infty }f_{n}(x_{n})\leq f(x){\text{ for every }}x_{n}\to x}$

and

${\displaystyle \liminf _{n\to \infty }f_{n}(x_{n})\geq f(x){\text{ for some }}x_{n}\to x.}$

Relationship to minimization problems

Assume we have a difficult minimization problem

${\displaystyle \inf _{x\in C}g(x)}$

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

${\displaystyle \inf _{x\in C_{n}}g_{n}(x)}$

for functions ${\displaystyle g_{n}}$ and sets ${\displaystyle C_{n}}$.

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

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

So that the problems ${\displaystyle \inf _{x\in X}f(x)}$ and ${\displaystyle \inf _{x\in X}f_{n}(x)}$ are equivalent to the original and approximate problems, respectively.

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

${\displaystyle \lim _{n\to \infty }\operatorname {argmin} f_{n}\subseteq \operatorname {argmin} f.}$

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

Properties

• ${\displaystyle (f_{n})}$ epi-converges to ${\displaystyle f}$ if and only if ${\displaystyle (-f_{n})}$ hypo-converges to ${\displaystyle -f}$.
• ${\displaystyle (f_{n})}$ epi-converges to ${\displaystyle f}$ if and only if ${\displaystyle (\operatorname {epi} f_{n})}$ converges to ${\displaystyle \operatorname {epi} f}$ as sets, in the Painlevé–Kuratowski sense of set convergence. Here, ${\displaystyle \operatorname {epi} f}$ is the epigraph of the function ${\displaystyle f}$.
• If ${\displaystyle f_{n}}$ epi-converges to ${\displaystyle f}$, then ${\displaystyle f}$ is lower semi-continuous.
• If ${\displaystyle f_{n}}$ is convex for each ${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle (f_{n})}$ epi-converges to ${\displaystyle f}$, then ${\displaystyle f}$ is convex.
• If ${\displaystyle f_{n}^{1}\leq f_{n}\leq f_{n}^{2}}$ and both ${\displaystyle (f_{n}^{1})}$ and ${\displaystyle (f_{n}^{2})}$ epi-converge to ${\displaystyle f}$, then ${\displaystyle (f_{n})}$ epi-converges to ${\displaystyle f}$.
• If ${\displaystyle (f_{n})}$ converges uniformly to ${\displaystyle f}$ on each compact set of ${\displaystyle \mathbb {R} _{n}}$ and ${\displaystyle (f_{n})}$ are continuous, then ${\displaystyle (f_{n})}$ epi-converges and hypo-converges to ${\displaystyle f}$.
• 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. Grundlehren der mathematischen Wissenschaften. Vol. 317. Springer Science & Business Media. pp. 238–297. doi:10.1007/978-3-642-02431-3_7. ISBN   978-3-540-62772-2.
• 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 C. 6: 73–100. Bibcode:1989AIHPC...6...73A. doi:10.1016/S0294-1449(17)30036-7.