Linear matrix inequality

In <a href="/facts/Convex_optimization/7D6U4MTN">convex optimization</a>, a linear matrix inequality (LMI) is an expression of the form

LMI
        ⁡
        (
        y
        )
        :=
        
          A
          
            0
          
        
        +
        
          y
          
            1
          
        
        
          A
          
            1
          
        
        +
        
          y
          
            2
          
        
        
          A
          
            2
          
        
        +
        ⋯
        +
        
          y
          
            m
          
        
        
          A
          
            m
          
        
        ⪰
        0
        
      
    
    {\displaystyle \operatorname {LMI} (y):=A_{0}+y_{1}A_{1}+y_{2}A_{2}+\cdots +y_{m}A_{m}\succeq 0\,}

where

<ul><li>
 
 
 
 y
 =
 [
 
 y
 
 i
 
 
 
 ,
  
 i
 
 =
 
 1
 ,
 …
 ,
 m
 ]
 
 
 {\displaystyle y=[y_{i}\,,~i\!=\!1,\dots ,m]}
 
 is a real vector,</li>
<li>
 
 
 
 
 A
 
 0
 
 
 ,
 
 A
 
 1
 
 
 ,
 
 A
 
 2
 
 
 ,
 …
 ,
 
 A
 
 m
 
 
 
 
 {\displaystyle A_{0},A_{1},A_{2},\dots ,A_{m}}
 
 are 
 
 
 
 n
 ×
 n
 
 
 {\displaystyle n\times n}
 
 <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrices</a> 
 
 
 
 
 
 S
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {S} ^{n}}
 
,</li>
<li>
 
 
 
 B
 ⪰
 0
 
 
 {\displaystyle B\succeq 0}
 
 is a generalized inequality meaning 
 
 
 
 B
 
 
 {\displaystyle B}
 
 is a <a href="/facts/Positive_semidefinite_matrix/Hbr6AuPS">positive semidefinite matrix</a> belonging to the positive semidefinite cone 
 
 
 
 
 
 S
 
 
 +
 
 
 
 
 {\displaystyle \mathbb {S} _{+}}
 
 in the subspace of symmetric matrices 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
.</li></ul>
This linear matrix inequality specifies a <a href="/facts/Convex_set/vdAuJRJl">convex</a> constraint on 
 
 
 
 y
 
 
 {\displaystyle y}
 
.

Linear matrix inequality open-in-new

Linear matrix inequality