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 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

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
In fact, epi-convergence coincides with the -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. hypo-converges to if
and

Relationship to minimization problems

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