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| In [[convex optimization]], a '''linear matrix inequality''' ('''LMI''') is an expression of the form
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| : <math>\operatorname{LMI}(y):=A_0+y_1A_1+y_2A_2+\cdots+y_m A_m\geq0\,</math>
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| where
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| * <math>y=[y_i\,,~i\!=\!1,\dots, m]</math> is a real vector,
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| * <math>A_0, A_1, A_2,\dots,A_m</math> are <math>n\times n</math> [[symmetric matrix | symmetric matrices]] <math>\mathbb{S}^n</math>,
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| * <math>B\geq0 </math> is a generalized inequality meaning <math>B</math> is a [[positive semidefinite matrix]] belonging to the positive semidefinite cone <math>\mathbb{S}_+</math> in the subspace of symmetric matrices <math>\mathbb{S}</math>.
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| This linear matrix inequality specifies a [[convex set|convex]] constraint on ''y''.
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| == Applications ==
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| 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.
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| Many optimization problems in [[control theory]], [[system identification]] and [[signal processing]] can be formulated using LMIs. Also LMIs find application in [[Polynomial SOS|Polynomial Sum-Of-Squares]]. The prototypical primal and dual [[semidefinite programming|semidefinite program]] is a minimization of a real linear function respectively subject to the primal and dual [[convex cone]]s governing this LMI.
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| == Solving LMIs ==
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| A major breakthrough in convex optimization lies in the introduction of [[interior-point method]]s. 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.
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| == References ==
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| * Y. Nesterov and A. Nemirovsky, ''Interior Point Polynomial Methods in Convex Programming.'' SIAM, 1994.
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| == External links ==
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| * S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, [http://www.stanford.edu/~boyd/lmibook/ Linear Matrix Inequalities in System and Control Theory] (book in pdf)
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| * C. Scherer and S. Weiland [http://www.dcsc.tudelft.nl/~cscherer/lmi.html Course on Linear Matrix Inequalities in Control], Dutch Institute of Systems and Control (DISC).
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| [[Category:Mathematical optimization]]
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