Epi-convergence

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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[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]