Linear Matrix Inequalities (LMIs)
A linear matrix inequality (LMI) is an expression of the form
$$A_0 + y_1 A_1 + y_2 A_2 + \cdots + y_m A_m \succeq O_n$$
where $y \in \mathbb R^m$ and $A_0, A_1, \dots, A_m$ are symmetric $n \times n$ matrices. The generalized inequality $Q \succeq O_n$ means that $Q$ is a positive semidefinite matrix.
This linear matrix inequality specifies a convex constraint on $y$.
There are efficient numerical methods to determine whether an LMI is feasible (e.g., whether there exists a vector $y$ such that LMI($y) \ge 0$), 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.
See also: Wikipedia
