Concept

# Linear matrix inequality

Summary
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
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