Linear matrix inequality

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

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

where

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

This linear matrix inequality specifies a convex constraint on ${\displaystyle y}$.

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.

• Semidefinite programming
• Spectrahedron
• Finsler's lemma

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

• 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