Linear matrix inequality

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

$\operatorname {LMI} (y):=A_{0}+y_{1}A_{1}+y_{2}A_{2}+\cdots +y_{m}A_{m}\succeq 0\,$

where

• $y=[y_{i}\,,~i\!=\!1,\dots ,m]$ is a real vector,
• $A_{0},A_{1},A_{2},\dots ,A_{m}$ are $n\times n$ symmetric matrices $\mathbb {S} ^{n}$,
• $B\succeq 0$ is a generalized inequality meaning $B$ is a positive semidefinite matrix belonging to the positive semidefinite cone $\mathbb {S} _{+}$ 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