# Linear matrix inequality

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

$\operatorname{LMI}(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\geq0\,$

where

• $y=[y_i\,,~i\!=\!1,\dots, m]$ is a real vector,
• $A_0, A_1, A_2,\dots,A_m$ are $n\times n$ symmetric matrices $\mathbb{S}^n$,
• $B\geq0$ is a generalized inequality meaning $B$ is a positive semidefinite matrix belonging to the positive semidefinite cone $\mathbb{S}_+$ in the subspace of symmetric matrices $\mathbb{S}$.

This linear matrix inequality specifies a convex constraint on y.

## Applications

There are efficient numerical methods to determine whether an LMI is feasible (e.g., whether there exists a vector y such that LMI(y) ≥ 0), or to solve a convex optimization problem with LMI constraints. Many optimization problems in control theory, system identification and signal processing can be formulated using LMIs. Also LMIs find application in Polynomial Sum-Of-Squares. The prototypical primal and dual semidefinite program is a minimization of a real linear function respectively subject to the primal and dual convex cones governing this LMI.

## Solving LMIs

A major breakthrough in convex optimization lies in the introduction of interior-point methods. These methods were developed in a series of papers and became of true interest in the context of LMI problems in the work of Yurii Nesterov and Arkadii Nemirovskii.

## References

• Y. Nesterov and A. Nemirovsky, Interior Point Polynomial Methods in Convex Programming. SIAM, 1994.