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.
 Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a metric space, and 
  
    
      
        
          f
          
            n
          
        
        :
        X
        →
        
          R
        
      
    
    {\displaystyle f_{n}:X\to \mathbb {R} }
  
 a real-valued function for each natural number 
  
    
      
        n
      
    
    {\displaystyle n}
  
. We say that the sequence 
  
    
      
        (
        
          f
          
            n
          
        
        )
      
    
    {\displaystyle (f^{n})}
  
 epi-converges to a function 
  
    
      
        f
        :
        X
        →
        
          R
        
      
    
    {\displaystyle f:X\to \mathbb {R} }
  
 if for each 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
  

 The following extension allows epi-convergence to be applied to a sequence of functions with non-constant domain.
 Denote by 
  
    
      
        
          
            
              R
            
            ¯
          
        
        =
        
          R
        
        ∪
        {
        ±
        ∞
        }
      
    
    {\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{\pm \infty \}}
  
 the extended real numbers. Let 
  
    
      
        
          f
          
            n
          
        
      
    
    {\displaystyle f_{n}}
  
 be a function 
  
    
      
        
          f
          
            n
          
        
        :
        X
        →
        
          
            
              R
            
            ¯
          
        
      
    
    {\displaystyle f_{n}:X\to {\overline {\mathbb {R} }}}
  
 for each 
  
    
      
        n
        ∈
        
          N
        
      
    
    {\displaystyle n\in \mathbb {N} }
  
. The sequence 
  
    
      
        (
        
          f
          
            n
          
        
        )
      
    
    {\displaystyle (f_{n})}
  
 epi-converges to 
  
    
      
        f
        :
        X
        →
        
          
            
              R
            
            ¯
          
        
      
    
    {\displaystyle f:X\to {\overline {\mathbb {R} }}}
  
 if for each 
  
    
      
        x
        ∈
        X
      
    
    {\displaystyle x\in X}
  

 In fact, epi-convergence coincides with the 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
-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. 
  
    
      
        (
        
          f
          
            n
          
        
        )
      
    
    {\displaystyle (f_{n})}
  
 hypo-converges to 
  
    
      
        f
      
    
    {\displaystyle f}
  
 if
 and
 Assume we have a difficult minimization problem 
 where 
  
    
      
        g
        :
        X
        →
        
          R
        
      
    
    {\displaystyle g:X\to \mathbb {R} }
  
 and 
  
    
      
        C
        ⊆
        X
      
    
    {\displaystyle C\subseteq X}
  
. We can attempt to approximate this problem by a sequence of easier problems
 for functions 
  
    
      
        
          g
          
            n
          
        
      
    
    {\displaystyle g_{n}}
  
 and sets 
  
    
      
        
          C
          
            n
          
        
      
    
    {\displaystyle C_{n}}
  
.
 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
          
            x
            ∈
            X
          
        
        f
        (
        x
        )
      
    
    {\displaystyle \inf _{x\in X}f(x)}
  
 and 
  
    
      
        
          inf
          
            x
            ∈
            X
          
        
        
          f
          
            n
          
        
        (
        x
        )
      
    
    {\displaystyle \inf _{x\in X}f_{n}(x)}
  
 are equivalent to the original and approximate problems, respectively.
 If 
  
    
      
        (
        
          f
          
            n
          
        
        )
      
    
    {\displaystyle (f_{n})}
  
 epi-converges to 
  
    
      
        f
      
    
    {\displaystyle f}
  
, then 
  
    
      
        
          lim sup
          
            n
            →
            ∞
          
        
        [
        inf
        
          f
          
            n
          
        
        ]
        ≤
        inf
        f
      
    
    {\displaystyle \limsup _{n\to \infty }[\inf f_{n}]\leq \inf f}
  
. Furthermore, if 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is a limit point of minimizers of 
  
    
      
        
          f
          
            n
          
        
      
    
    {\displaystyle f_{n}}
  
, then 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is a minimizer of 
  
    
      
        f
      
    
    {\displaystyle f}
  
. In this sense,
 Epi-convergence is the weakest notion of convergence for which this result holds.