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Concept
Linear matrix inequality
Summary
In convex optimization, a linear matrix inequality (LMI) 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 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) ≥ 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.
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.
Official source
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