Linear matrix inequality

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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]
* S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, [http://www.stanford.edu/~boyd/lmibook/ Linear Matrix Inequalities in System and Control Theory]
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* 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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* C. Scherer and S. Weiland [http://www.dcsc.tudelft.nl/~cscherer/2416/lmi.html Course on Linear Matrix Inequalities in Control], Dutch Institute of Systems and Control (DISC).

Revision as of 18:24, 6 April 2009

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

LaTeX: LMI(y):=A_0+y_1A_1+y_2A_2+\dots+y_m A_m\succeq0\,

where

  • LaTeX: y=[y_i\,,~i\!=\!1\dots m] is a real vector,
  • LaTeX: A_0\,, A_1\,, A_2\,,\dots\,A_m are symmetric matrices in the subspace of LaTeX: n\times n symmetric matrices LaTeX: \mathbb{S}^n,
  • LaTeX: B\succeq0 is a generalized inequality meaning LaTeX: B is a positive semidefinite matrix belonging to the positive semidefinite cone LaTeX: \mathbb{S}_+ in the subspace of symmetric matrices LaTeX: \mathbb{S}.

This linear matrix inequality specifies a convex constraint on y.

Contents

Convexity of the LMI constraint

LaTeX: LMI(y)\succeq 0 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, LaTeX: x\succeq0):

for LaTeX: X\!\in\mathbb{S}^n given LaTeX: \,A_j\!\in\mathbb{S}^n, LaTeX: \,j\!=\!1\ldots m

LaTeX: \begin{array}{ll}\mathcal{K}
\!\!&=\left\{\left[\begin{array}{c}\langle A_1\,,\,X^{}\rangle\\\vdots\\\langle A_m\;,\,X^{}\rangle\end{array}\right]|~X\!\succeq_{\!}0\right\}\subseteq_{}\reals^m\\\\
&=\left\{\left[\begin{array}{c}\textrm{svec}(A_1)^T\\\vdots\\\textrm{svec}(A_m)^T\end{array}\right]\!\textrm{svec}X~|~X\!\succeq_{\!}0\right\}\\\\
&:=\;\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0_{}\}
\end{array}

where

  • LaTeX: A\!\in_{}\!\mathbb{R}^{m\times n(n+1)/2},
  • symmetric vectorization svec is a stacking of columns defined in (Dattorro, Ch.2.2.2.1),
  • LaTeX: A_0=\mathbf{0} is assumed without loss of generality.

LaTeX: \mathcal{K} is a convex cone because

LaTeX: A\,\textrm{svec}{X_{{\rm p}_1}}_{\,},_{_{}}A\,\textrm{svec}{X_{{\rm p}_2}}\!\in\mathcal{K}~\Rightarrow~
A(\zeta_{\,}\textrm{svec}{X_{{\rm p}_1\!}}+_{}\xi_{\,}\textrm{svec}{X_{{\rm p}_2}})\in_{}\mathcal{K}
\textrm{~~for\,all~\,}\zeta_{\,},\xi\geq0

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

Now consider the (closed convex) dual cone:

LaTeX: \begin{array}{rl}\mathcal{K}^*
\!\!\!&=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z\!\in_{_{}\!}\mathcal{K}_{}\right\}\subseteq_{}\reals^m\\
&=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z_{\!}=_{\!}A\,\textrm{svec}X\,,~X\succeq0_{}\right\}\\
&=_{}\left\{_{}y~|~\langle A\,\textrm{svec}X\,,~y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,X\!\succeq_{_{}\!}0_{}\right\}\\
&=\left\{y~|~\langle\textrm{svec}X\,,\,A^{T\!}y\rangle\geq_{}0\;~\textrm{for\,all}~\,X\!\succeq_{\!}0\right\}\\
&=\left\{y~|~\textrm{svec}^{-1}(A^{T\!}y)\succeq_{}0\right\}
\end{array}

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

  • LaTeX: Y\succeq0~\Leftrightarrow~\langle Y\,,\,X\rangle\geq0\;~\textrm{for\,all}~\,X\succeq0

This leads directly to an equally peculiar halfspace-description

LaTeX: \mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0_{}\}

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

LMI Geometry

Although matrix LaTeX: \,A\, is finite-dimensional, LaTeX: \mathcal{K} is generally not a polyhedral cone (unless LaTeX: \,m\, equals 1 or 2) simply because LaTeX: \,X\!\in\mathbb{S}_+^n\,.

Provided the LaTeX: A_j matrices are linearly independent, then relative interior = interior

LaTeX: \textrm{rel\,int}\mathcal{K}=\textrm{int}\mathcal{K}

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

If matrix LaTeX: \,A\, has no nullspace, on the other hand, then LaTeX: \,A\,\textrm{svec}X\, is an isomorphism in LaTeX: \,X\, between the positive semidefinite cone LaTeX: \mathbb{S}_+^n and range LaTeX: \,\mathcal{R}(A)\, of matrix LaTeX: \,A.

In that case, convex cone LaTeX: \,\mathcal{K}\, has relative interior

LaTeX: \textrm{rel\,int}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succ_{\!}0_{}\}

and boundary

LaTeX: \textrm{rel}\,\partial^{}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0\,,~X\!\nsucc_{\!}0_{}\}


When the LaTeX: A_j matrices are linearly independent, function LaTeX: \,g(y)_{\!}:=_{_{}\!}\sum y_jA_j\, on LaTeX: \mathbb{R}^m is a linear bijection.

Inverse image of the positive semidefinite cone under LaTeX: \,g(y)\, must therefore have dimension LaTeX: _{}m .

In that circumstance, the dual cone interior is nonempty

LaTeX: \textrm{int}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succ_{}0_{}\}

having boundary

LaTeX: \partial^{}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0\,,~\sum\limits_{j=1}^my_jA_j\nsucc_{}0_{}\}

Applications

There are efficient numerical methods to determine whether an LMI is feasible (i.e., whether there exists a vector LaTeX: y such that LaTeX: LMI(y)\succeq0 ), 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 are optimizations of a real linear function respectively subject to the primal and dual convex cones governing this LMI.

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