- Linear matrix inequality
In
convex optimization , a linear matrix inequality (LMI) is an expression of the form: LMI(y):=A_0+y_1A_1+y_2A_2+cdots+y_m A_mgeq0,where
* y= [y_i,,~i!=!1dots m] is a real vector,
* A_0,, A_1,, A_2,,dots,A_m are symmetric matrices in the subspace of n imes n symmetric matrices mathbb{S}^n,
* Bgeq0 is a generalized inequality meaning B is apositive 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 ("i.e.", whether there exists a vector y such that LMI(y)geq0 ), or to solve a
convex optimization problem with LMI constraints.Many optimization problems incontrol theory ,system identification andsignal processing can be formulated using LMIs. Also LMIs find application inPolynomial SOS . The prototypical primal and dual semidefinite program is a minimization of a real linear function respectively subject to the primal and dualconvex cone s governing this LMI.Solving LMIs
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.References
* Y. Nesterov and A. Nemirovsky, "Interior Point Polynomial Methods in Convex Programming." SIAM, 1994.
*. Chapter 2 explains cones and their duals.
External links
* 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)
* C. Scherer and S. Weiland [http://www.cs.ele.tue.nl/sweiland/lmi.html Course on Linear Matrix Inequalities in Control] , Dutch Institute of Systems and Control (DISC).
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