Linear matrix inequality explained

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

\operatorname{LMI}(y): = A _{0+y} _{1A} _{1+y} _{2A} _{2+ … +y} _{m} A _{m\succeq} 0 wherey = [y _{i,~i=1,...,} m] is a real vector,A _{0,} A _{1,} A _{2,...,A} _{m} are n x n symmetric matrices S ^{n} ,B\succeq 0 is a generalized inequality meaning B is a positive semidefinite matrix belonging to the positive semidefinite cone S _{+} in the subspace of symmetric matrices 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