Linear matrix inequality

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

[math]\displaystyle{ \operatorname{LMI}(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\succeq 0\, }[/math]

where

• [math]\displaystyle{ y=[y_i\,,~i\!=\!1,\dots, m] }[/math] is a real vector,
• [math]\displaystyle{ A_0, A_1, A_2,\dots,A_m }[/math] are [math]\displaystyle{ n\times n }[/math] symmetric matrices [math]\displaystyle{ \mathbb{S}^n }[/math],
• [math]\displaystyle{ B\succeq0 }[/math] is a generalized inequality meaning [math]\displaystyle{ B }[/math] is a positive semidefinite matrix belonging to the positive semidefinite cone [math]\displaystyle{ \mathbb{S}_+ }[/math] in the subspace of symmetric matrices [math]\displaystyle{ \mathbb{S} }[/math].

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

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