Linear matrix inequality

In convex optimization, a linear matrix inequality is an expression of the form
where

is a real vector,
are symmetric matrices,
is a generalized inequality meaning is a positive semidefinite matrix belonging to the positive semidefinite cone in the subspace of symmetric matrices.

This linear matrix inequality specifies a convex constraint on .

Applications

There are efficient numerical methods to determine whether an LMI is feasible, 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.