Linear Matrix Inequalities

In convex optimization, a linear matrix inequality (LMI) is an expression of the form
: $\operatorname(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\succeq 0\,$
where
* y= _i\,,~i\!=\!1,\dots, m/math> is a real vector,
* $A_0, A_1, A_2,\dots,A_m$ are $n\times n$ symmetric matrices $\mathbb^n$,
* $B\succeq0$ is a generalized inequality meaning $B$ is a positive semidefinite matrix belonging to the positive semidefinite cone $\mathbb_+$ in the subspace of symmetric matrices $\mathbb{S}$.

This linear matrix inequality specifies a convex constraint on $y$.

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

Convex optimization