Linear matrix inequality

Short description: Mathematical convex optimization

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

LMI⁡(y):=A0+y1A1+y2A2+⋯+ymAm⪰0

where

* y=[yi,i=1,…,m] is a real vector,
* A0,A1,A2,…,Am are n×n symmetric matrices 𝕊n,
* B⪰0 is a generalized inequality meaning B is a positive semidefinite matrix belonging to the positive semidefinite cone 𝕊+ in the subspace of symmetric matrices 𝕊.

This linear matrix inequality specifies a convex constraint on y.

## Contents

* 1 Applications
* 2 Solving LMIs
* 3 See also
* 4 References
* 5 External links

## 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.

## See also

* Semidefinite programming
* Spectrahedron
* Finsler's lemma

## References

* Y. Nesterov and A. Nemirovsky, Interior Point Polynomial Methods in Convex Programming. SIAM, 1994.

## External links

* S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (book in pdf)
* C. Scherer and S. Weiland, Linear Matrix Inequalities in Control

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