Riemann Xi function

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

$\mathrm{LMI}(y):=A_0+y_1{A}_1+y_2{A}_2+\cdots +y_m{A}_m\ge 0$

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 ${\mathbb{𝕊}}^n$,
• $B\ge 0$ 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 lies in 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 Arkadii Nemirovskii.

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 Course on Linear Matrix Inequalities in Control, Dutch Institute of Systems and Control (DISC).