In convex optimization, 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

• y = [ y i ,   i = 1 , … , m ] {\displaystyle y=[y_{i}\,,~i\!=\!1,\dots ,m]} is a real vector,
• 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} symmetric matrices S n {\displaystyle \mathbb {S} ^{n}} ,
• B ⪰ 0 {\displaystyle B\succeq 0} is a generalized inequality meaning B {\displaystyle B} is a positive semidefinite matrix belonging to the positive semidefinite cone S + {\displaystyle \mathbb {S} _{+}} in the subspace of symmetric matrices S {\displaystyle \mathbb {S} } .

This linear matrix inequality specifies a convex constraint on  y {\displaystyle y} .