Linear matrix inequality

In convex optimization, a linear matrix inequality (LMI) is an expression of the form
 * $$\operatorname{LMI}(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\succeq 0\,$$

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 $$\mathbb{S}^n$$,
 * $$B\succeq0 $$ 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 was 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 Arkadi Nemirovski.