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.
Let X be a metric space, and f ν : X → R a real-valued function for each natural number ν . We say that the sequence ( f ν ) epi-converges to a function f : X → R if for each x ∈ X
lim inf ν → ∞ f ν ( x ν ) ≥ f ( x ) for every x ν → x and lim sup ν → ∞ f ν ( x ν ) ≤ f ( x ) for some x ν → x . The following extension allows epi-convergence to be applied to a sequence of functions with non-constant domain.
Denote by R ¯ = R ∪ { ± ∞ } the extended real numbers. Let f ν be a function f ν : X → R ¯ for each ν ∈ N . The sequence ( f ν ) epi-converges to f : X → R ¯ if for each x ∈ X
lim inf ν → ∞ f ν ( x ν ) ≥ f ( x ) for every x ν → x and lim sup ν → ∞ f ν ( x ν ) ≤ f ( x ) for some x ν → x . Epi-convergence is the appropriate topology with which to approximate minimization problems. For maximization problems one uses the symmetric notion of hypo-convergence. ( f ν ) hypo-converges to f if
lim sup ν → ∞ f ν ( x ν ) ≤ f ( x ) for every x ν → x and
lim inf ν → ∞ f ν ( x ν ) ≥ f ( x ) for some x ν → x . Assume we have a difficult minimization problem
inf x ∈ C g ( x ) where g : X → R and C ⊆ X . We can attempt to approximate this problem by a sequence of easier problems
inf x ∈ C ν g ν ( x ) for functions g ν and sets C ν .
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
f ( x ) = { g ( x ) , x ∈ C , ∞ , x ∉ C , f ν ( x ) = { g ν ( x ) , x ∈ C ν , ∞ , x ∉ C ν . So that the problems inf x ∈ X f ( x ) and inf x ∈ X f ν ( x ) are equivalent to the original and approximate problems, respectively.
If ( f ν ) epi-converges to f , then lim sup ν → ∞ [ inf f ν ] ≤ inf f . Furthermore, if x is a limit point of minimizers of f ν , then x is a minimizer of f . In this sense,
lim v → ∞ argmin f ν ⊆ argmin f . Epi-convergence is the weakest notion of convergence for which this result holds.
( f ν ) epi-converges to f if and only if ( − f ν ) hypo-converges to − f . ( f ν ) epi-converges to f if and only if ( epi f ν ) converges to epi f as sets, in the Painlevé–Kuratowski sense of set convergence. Here, epi f is the epigraph of the function f .If f ν epi-converges to f , then f is lower semi-continuous.If f ν is convex for each ν ∈ N and ( f ν ) epi-converges to f , then f is convex.If f 1 ν ≤ f ν ≤ f 2 ν and both ( f 1 ν ) and ( f 2 ν ) epi-converge to f , then ( f ν ) epi-converges to f .If ( f ν ) converges uniformly to f on each compact set of R n , then ( f ν ) 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.