Linear matrix inequality

From Wikimization

Jump to: navigation, search

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

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

where

  • y=[y_i\,,~i\!=\!1\dots m] is a real vector,
  • A_0\,, A_1\,, A_2\,,\dots\,A_m are symmetric matrices in the subspace of 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.

Contents

Convexity of the LMI constraint

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 convex cone defined over the positive semidefinite cone

(instead of the more common nonnegative orthant, x\succeq0):

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

\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

  • 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),
  • A_0=\mathbf{0} is assumed without loss of generality.

\mathcal{K} is a convex cone because

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:

\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:

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

This leads directly to an equally peculiar halfspace-description

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

Relative interior of \mathcal{K} may always be expressed \textrm{rel\,int}\,\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succ_{\!}0_{}\}.

Provided the \,A_j matrices are linearly independent, then \textrm{rel\,int}\,\mathcal{K}=\textrm{int}\,\mathcal{K}

meaning, cone \mathcal{K} interior is nonempty; implying, dual cone \mathcal{K}^* is pointed (Dattorro, ch.2).

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

That is sufficient for convex cone \,\mathcal{K}\, to be closed, and necessary to have relative boundary \textrm{rel}\,\partial^{}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0\,,~X\!\nsucc_{\!}0_{}\}.


Relative interior of the dual cone may always be expressed \textrm{rel\,int}\,\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succ_{}0_{}\}.

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

Inverse image of the positive semidefinite cone under \,g(y)\, must therefore have dimension equal to \dim\!\left(\mathcal{R}(A^{\rm T})_{}\!\cap\mbox{svec}\,\mathbb{S}_+^{_{}n}\right)

and relative boundary \textrm{rel\,}\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_{}\}.

When this dimension is \,m\,, the dual cone interior is nonempty \textrm{rel\,int}\,\mathcal{K}^*=\textrm{int}\,\mathcal{K}^*

and closure of convex cone \mathcal{K} is pointed.

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)\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.

External links

Retrieved from "http://www.convexoptimization.com/wikimization/index.php/Linear_matrix_inequality"
Personal tools