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.
be a metric space, and
a real-valued function for each natural number
We say that the sequence
epi-converges to a function
The following extension allows epi-convergence to be applied to a sequence of functions with non-constant domain.
the extended real numbers.
The sequence
In fact, epi-convergence coincides with the
-convergence in first countable spaces.
Epi-convergence is the appropriate topology with which to approximate minimization problems.
For maximization problems one uses the symmetric notion of hypo-convergence.
if and Assume we have a difficult minimization problem where
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
inf
inf
are equivalent to the original and approximate problems, respectively.
lim sup
[ inf
] ≤ inf f
is a limit point of minimizers of
In this sense, Epi-convergence is the weakest notion of convergence for which this result holds.