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Linear matrix inequality

Mathematical convex optimization

Summary

Mathematical convex optimization

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+\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\succeq0 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.

References

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