# 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.

## Contents

## Definition[edit]

Let be a metric space, and a real-valued function for each natural number . We say that the sequence **epi-converges** to a function if for each

### Extended real-valued extension[edit]

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

Denote by the extended real numbers. Let be a function for each . The sequence epi-converges to if for each

### Hypo-convergence[edit]

Epi-convergence is the appropriate topology with which to approximate minimization problems. For maximization problems one uses the symmetric notion of **hypo-convergence**. hypo-converges to if

and

## Relationship to minimization problems[edit]

Assume we have a difficult minimization problem

where and . We can attempt to approximate this problem by a sequence of easier problems

for functions and sets .

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

So that the problems and are equivalent to the original and approximate problems, respectively.

If epi-converges to , then . Furthermore, if is a limit point of minimizers of , then is a minimizer of . In this sense,

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

## Properties[edit]

- epi-converges to if and only if hypo-converges to .
- epi-converges to if and only if converges to as sets, in the Painlevé–Kuratowski sense of set convergence. Here, is the epigraph of the function .
- If epi-converges to , then is lower semi-continuous.
- If is convex for each and epi-converges to , then is convex.
- If and both and epi-converge to , then epi-converges to .
- If converges uniformly to on each compact set of , then epi-converges and hypo-converges to .
- 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[edit]

- R. Tyrrell Rockafellar and Roger Wets, Variational Analysis. Chapter 7 ; Vol. 317. Springer Science & Business Media, 2009.
- Peter Kall, Approximation to optimization problems: An elementary review; Mathematics of Operations Research
**19**, pp. 9–18 (1986) - Hedy Attouch and Roger Wets, Epigraphical analysis; Annales de l'IHP Analyse non linéaire Vol. 6. 1989.