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I have a question about epi-converge as defined in https://en.wikipedia.org/wiki/Epi-convergence.
where $\liminf_{v \to \infty}f^v(x^v)\geq f(x)$ and $\limsup_{v \to \infty}f^v(x^v)\leq f(x)$
But $\inf\{\} ≤ \sup\{\}$, how can the above set of inequality stand?

Thanks

1 Answers 1

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You did not reproduce the full definition. $$ \liminf_{v \to \infty}f^v(x^v)\geq f(x) $$ has to hold for every sequence $x^v$ while $$ \limsup_{v \to \infty}f^v(x^v)\leq f(x) $$ has to hold for at least one sequence. For those sequences where the second condition holds, you have of course in combination that $$\lim_{v \to \infty}f^v(x^v)= f(x).$$