Linear matrix inequality

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

$A(x) := A_0 + x_1 A_1 + x_2 A_2 + \dots + x_n A_n \leq 0 \,$

where

$x = (x_1, \dots, x_n)$ is a real vector,
$A_0, A_1, A_2, \dots, A_n$ are symmetric matrices,
the $A \leq 0$ inequality means that A is a negative-semidefinite matrix.

This inequality defines a convex constraint on x. There are efficient numerical methods to determine whether an LMI is feasible (i.e., whether there exists an x such that $A(x) \leq 0$ ), or to solve a convex optimization problem with LMI constraints.

1 Applications
2 Solving LMIs
3 References
4 External links

[edit] Applications

Many optimization problems in control theory, system identification and signal processing can be formulated using LMIs. A semidefinite program is the minimization of a linear functional subject to an LMI.

[edit] 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.