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Linear matrix inequality - Wikipedia

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Mathematical convex optimization

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

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

where

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

This linear matrix inequality specifies a convex constraint on  y {\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

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