Linear matrix inequality

In convex optimization, a linear matrix inequality (LMI) is an expression of the form

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

where

• ${\displaystyle y=[y_{i}\,,~i\!=\!1,\dots ,m]}$ is a real vector,
• ${\displaystyle A_{0},A_{1},A_{2},\dots ,A_{m}}$ are ${\displaystyle n\times n}$ symmetric matrices ${\displaystyle \mathbb {S} ^{n}}$,
• ${\displaystyle B\succeq 0}$ is a generalized inequality meaning ${\displaystyle B}$ is a positive semidefinite matrix belonging to the positive semidefinite cone ${\displaystyle \mathbb {S} _{+}}$ in the subspace of symmetric matrices ${\displaystyle \mathbb {S} }$.

This linear matrix inequality specifies a convex constraint on y.