Characteristic function (convex analysis)

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>, and let 
  
    
      
        A
      
    
    {\displaystyle A}
  
 be a <a href="/facts/Subset/SJGERmbA">subset</a> of 
  
    
      
        X
      
    
    {\displaystyle X}
  
. The characteristic function of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the function
</p>

χ
          
            A
          
        
        :
        X
        →
        
          R
        
        ∪
        {
        +
        ∞
        }
      
    
    {\displaystyle \chi _{A}:X\to \mathbb {R} \cup \{+\infty \}}

<p>taking values in the <a href="/facts/Extended_real_number_line/65Gs7QS0">extended real number line</a> defined by
</p>

χ
          
            A
          
        
        (
        x
        )
        :=
        
          
            {
            
              
                
                  0
                  ,
                
                
                  x
                  ∈
                  A
                  ;
                
              
              
                
                  +
                  ∞
                  ,
                
                
                  x
                  ∉
                  A
                  .
                
              
            
            
          
        
      
    
    {\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}

<h2 id="relationship-with-the-indicator-function">Relationship with the indicator function</h2>
<p>Let 
  
    
      
        
          
            1
          
          
            A
          
        
        :
        X
        →
        
          R
        
      
    
    {\displaystyle \mathbf {1} _{A}:X\to \mathbb {R} }
  
 denote the usual indicator function:
</p>

1
          
          
            A
          
        
        (
        x
        )
        :=
        
          
            {
            
              
                
                  1
                  ,
                
                
                  x
                  ∈
                  A
                  ;
                
              
              
                
                  0
                  ,
                
                
                  x
                  ∉
                  A
                  .
                
              
            
            
          
        
      
    
    {\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1,&x\in A;\\0,&x\not \in A.\end{cases}}}

<p>If one adopts the conventions that
</p>
<ul><li>for any 
  
    
      
        a
        ∈
        
          R
        
        ∪
        {
        +
        ∞
        }
      
    
    {\displaystyle a\in \mathbb {R} \cup \{+\infty \}}
  
, 
  
    
      
        a
        +
        (
        +
        ∞
        )
        =
        +
        ∞
      
    
    {\displaystyle a+(+\infty )=+\infty }
  
 and 
  
    
      
        a
        (
        +
        ∞
        )
        =
        +
        ∞
      
    
    {\displaystyle a(+\infty )=+\infty }
  
, except 
  
    
      
        0
        (
        +
        ∞
        )
        =
        0
      
    
    {\displaystyle 0(+\infty )=0}
  
;</li>
<li>
  
    
      
        
          
            1
            0
          
        
        =
        +
        ∞
      
    
    {\displaystyle {\frac {1}{0}}=+\infty }
  
; and</li>
<li>
  
    
      
        
          
            1
            
              +
              ∞
            
          
        
        =
        0
      
    
    {\displaystyle {\frac {1}{+\infty }}=0}
  
;</li></ul>
<p>then the indicator and characteristic functions are related by the equations
</p>

1
          
          
            A
          
        
        (
        x
        )
        =
        
          
            1
            
              1
              +
              
                χ
                
                  A
                
              
              (
              x
              )
            
          
        
      
    
    {\displaystyle \mathbf {1} _{A}(x)={\frac {1}{1+\chi _{A}(x)}}}

χ
          
            A
          
        
        (
        x
        )
        =
        (
        +
        ∞
        )
        
          (
          
            1
            −
            
              
                1
              
              
                A
              
            
            (
            x
            )
          
          )
        
        .
      
    
    {\displaystyle \chi _{A}(x)=(+\infty )\left(1-\mathbf {1} _{A}(x)\right).}

<h2 id="subgradient">Subgradient</h2>
<p>The subgradient of 
  
    
      
        
          χ
          
            A
          
        
        (
        x
        )
      
    
    {\displaystyle \chi _{A}(x)}
  
 for a set 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the <a href="/facts/Tangent_cone/PX8G7gnl">tangent cone</a> of that set in 
  
    
      
        x
      
    
    {\displaystyle x}
  
. 
</p>
<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/R._Tyrrell_Rockafellar/Jzj3n73g">Rockafellar, R. T.</a> (1997) [1970]. <i>Convex Analysis</i>. Princeton, NJ: Princeton University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-01586-6.</li></ul>

Characteristic function (convex analysis) open-in-new

Characteristic function (convex analysis)