# Linear matrix inequality

### From Wikimization

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== Convexity of the LMI constraint == | == Convexity of the LMI constraint == | ||

- | <math>LMI(y)\succeq 0</math> is a convex constraint on ''y'' which means membership to a dual (convex) cone as we now explain: '''('''[http://meboo.convexoptimization.com/ | + | <math>LMI(y)\succeq 0</math> is a convex constraint on ''y'' which means membership to a dual (convex) cone as we now explain: '''('''[http://meboo.convexoptimization.com/Meboo.html Dattorro, Example 2.13.5.1.1]''')''' |

Consider a peculiar vertex-description for a closed convex cone defined over the positive semidefinite cone | Consider a peculiar vertex-description for a closed convex cone defined over the positive semidefinite cone | ||

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where | where | ||

*<math>A\!\in_{}\!\mathbb{R}^{m\times n(n+1)/2}</math>, | *<math>A\!\in_{}\!\mathbb{R}^{m\times n(n+1)/2}</math>, | ||

- | *symmetric vectorization svec is a stacking of columns defined in '''('''[http://meboo.convexoptimization.com/ | + | *symmetric vectorization svec is a stacking of columns defined in '''('''[http://meboo.convexoptimization.com/Meboo.html Dattorro, Ch.2.2.2.1]''')''', |

*<math>A_0=\mathbf{0}</math> is assumed without loss of generality. | *<math>A_0=\mathbf{0}</math> is assumed without loss of generality. | ||

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<math>\textrm{rel\,int}\mathcal{K}=\textrm{int}\mathcal{K}</math> | <math>\textrm{rel\,int}\mathcal{K}=\textrm{int}\mathcal{K}</math> | ||

- | meaning, the cone interior is nonempty; implying, the dual cone is pointed ([http://meboo.convexoptimization.com/ | + | meaning, the cone interior is nonempty; implying, the dual cone is pointed ([http://meboo.convexoptimization.com/Meboo.html Dattorro, ch.2]). |

If matrix <math>\,A\,</math> has no nullspace, on the other hand, then | If matrix <math>\,A\,</math> has no nullspace, on the other hand, then |

## Revision as of 16:44, 22 February 2009

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

where

- is a real vector,
- are symmetric matrices in the subspace of symmetric matrices ,
- is a generalized inequality meaning is a positive semidefinite matrix belonging to the positive semidefinite cone in the subspace of symmetric matrices .

This linear matrix inequality specifies a convex constraint on *y*.

## Contents |

## Convexity of the LMI constraint

is a convex constraint on *y* which means membership to a dual (convex) cone as we now explain: **(**Dattorro, Example 2.13.5.1.1**)**

Consider a peculiar vertex-description for a closed convex cone defined over the positive semidefinite cone

**(**instead of the more common nonnegative orthant, **)**:

for given ,

where

- ,
- symmetric vectorization svec is a stacking of columns defined in
**(**Dattorro, Ch.2.2.2.1**)**, - is assumed without loss of generality.

is a convex cone because

since a nonnegatively weighted sum of positive semidefinite matrices must be positive semidefinite.

Now consider the (closed convex) dual cone:

that follows from Fejer's dual generalized inequalities for the positive semidefinite cone:

This leads directly to an equally peculiar halfspace-description

The summation inequality with respect to the positive semidefinite cone
is known as a *linear matrix inequality*.

## LMI Geometry

Although matrix is finite-dimensional, is generally not a polyhedral cone (unless equals 1 or 2) simply because .

Provided the matrices are linearly independent, then relative interior = interior

meaning, the cone interior is nonempty; implying, the dual cone is pointed (Dattorro, ch.2).

If matrix has no nullspace, on the other hand, then is an isomorphism in between the positive semidefinite cone and range of matrix .

In that case, convex cone has relative interior

and boundary

When the matrices are linearly independent, function on is a linear bijection.

Inverse image of the positive semidefinite cone under must therefore have dimension .

In that circumstance, the dual cone interior is nonempty

having boundary

## Applications

There are efficient numerical methods to determine whether an LMI is feasible (*i.e.*, whether there exists a vector such that ), 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. 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.

## External links

- S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory

- C. Scherer and S. Weiland Course on Linear Matrix Inequalities in Control, Dutch Institute of Systems and Control (DISC).