Linear matrix inequality

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Line 1: Line 1: In convex optimization, a '''linear matrix inequality (LMI)''' is an expression of the form In convex optimization, a '''linear matrix inequality (LMI)''' is an expression of the form - : $LMI(y):=A_0+y_1A_1+y_2A_2+\dots+y_m A_m\succeq0\,$ + : $LMI(y):=A_0+y_1A_1+y_2A_2+\ldots+y_m A_m\succeq0\,$ where where * $y=[y_i\,,~i\!=\!1\ldots m]$ is a real vector, * $y=[y_i\,,~i\!=\!1\ldots m]$ is a real vector, Line 18: Line 18: $\begin{array}{ll}\mathcal{K} [itex]\begin{array}{ll}\mathcal{K} - \!\!&=\left\{\left[\begin{array}{c}\langle A_1\,,\,X^{}\rangle\\\vdots\\\langle A_m\;,\,X^{}\rangle\end{array}\right]|~X\!\succeq_{\!}0\right\}\subseteq_{}\mathbb{R}^m\\\\ + \!\!&=\left\{\left[\begin{array}{c}\langle A_1\,,\,X^{}\rangle\\:\\\langle A_m\;,\,X^{}\rangle\end{array}\right]|~X\!\succeq_{\!}0\right\}\subseteq_{}\mathbb{R}^m\\\\ - &=\left\{\left[\begin{array}{c}\textrm{svec}(A_1)^T\\\vdots\\\textrm{svec}(A_m)^T\end{array}\right]\!\textrm{svec}X~|~X\!\succeq_{\!}0\right\}\\\\ + &=\left\{\left[\begin{array}{c}{\text svec}(A_1)^T\\:\\{\text svec}(A_m)^T\end{array}\right]{\text svec}X~|~X\!\succeq_{\!}0\right\}\\\\ - &:=\;\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0_{}\} + &:=\;\{A\,{\text svec}X~|~X\!\succeq_{\!}0_{}\} \end{array}$ \end{array}[/itex] Line 30: Line 30: $\mathcal{K}$ is a [[Convex cones|convex cone]] because $\mathcal{K}$ is a [[Convex cones|convex cone]] because - $A\,\textrm{svec}{X_{{\rm p}_1}}_{\,},_{_{}}A\,\textrm{svec}{X_{{\rm p}_2}}\!\in\mathcal{K}~\Rightarrow~ + [itex]A\,{\text svec}{X_{p_1}}_{\,},_{_{}}A\,{\text svec}{X_{p_2}}\!\in\mathcal{K}~\Rightarrow~ - A(\zeta_{\,}\textrm{svec}{X_{{\rm p}_1\!}}+_{}\xi_{\,}\textrm{svec}{X_{{\rm p}_2}})\in_{}\mathcal{K} + A(\zeta_{\,}{\text svec}{X_{p_1}}+_{}\xi_{\,}{\text svec}{X_{p_2}})\in_{}\mathcal{K} - \textrm{~~for\,all~\,}\zeta_{\,},\xi\geq0$ + {\text~~for\,all~\,}\zeta_{\,},\xi\geq0[/itex] since a nonnegatively weighted sum of positive semidefinite matrices must be positive semidefinite. since a nonnegatively weighted sum of positive semidefinite matrices must be positive semidefinite. Line 40: Line 40: $\begin{array}{rl}\mathcal{K}^* [itex]\begin{array}{rl}\mathcal{K}^* \!\!\!&=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z\!\in_{_{}\!}\mathcal{K}_{}\right\}\subseteq_{}\mathbb{R}^m\\ \!\!\!&=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z\!\in_{_{}\!}\mathcal{K}_{}\right\}\subseteq_{}\mathbb{R}^m\\ - &=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z_{\!}=_{\!}A\,\textrm{svec}X\,,~X\succeq0_{}\right\}\\ + &=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z_{\!}=_{\!}A\,{\text svec}X\,,~X\succeq0_{}\right\}\\ - &=_{}\left\{_{}y~|~\langle A\,\textrm{svec}X\,,~y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,X\!\succeq_{_{}\!}0_{}\right\}\\ + &=_{}\left\{_{}y~|~\langle A\,{\text svec}X\,,~y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,X\!\succeq_{_{}\!}0_{}\right\}\\ - &=\left\{y~|~\langle\textrm{svec}X\,,\,A^{T\!}y\rangle\geq_{}0\;~\textrm{for\,all}~\,X\!\succeq_{\!}0\right\}\\ + &=\left\{y~|~\langle{\text svec}X\,,\,A^{T\!}y\rangle\geq_{}0\;~\textrm{for\,all}~\,X\!\succeq_{\!}0\right\}\\ - &=\left\{y~|~\textrm{svec}^{-1}(A^{T\!}y)\succeq_{}0\right\} + &=\left\{y~|~{\text svec}^{-1}(A^{T\!}y)\succeq_{}0\right\} \end{array}$ \end{array}[/itex] Line 52: Line 52: This leads directly to an equally peculiar halfspace-description This leads directly to an equally peculiar halfspace-description - $\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0_{}\}$ + $\mathcal{K}^*=\{y\!\in\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0_{}\}$ The summation inequality with respect to the positive semidefinite cone The summation inequality with respect to the positive semidefinite cone Line 63: Line 63: Relative interior of $\mathcal{K}$ may always be expressed Relative interior of $\mathcal{K}$ may always be expressed - $\textrm{rel\,int}\,\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succ_{\!}0_{}\}.$ + $\textrm{rel\,int}\,\mathcal{K}=\{A\,{\text svec}X~|~X\!\succ0_{}\}.$ Provided the $\,A_j$ matrices are linearly independent, then Provided the $\,A_j$ matrices are linearly independent, then Line 71: Line 71: If matrix $\,A\,$ has no nullspace, then 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 $\,A.$ + $\,A\,{\text svec}X\,$ 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 cones|convex cone]] $\,\mathcal{K}\,$ to be closed, and necessary to have relative boundary That is sufficient for [[Convex cones|convex cone]] $\,\mathcal{K}\,$ to be closed, and necessary to have relative boundary - $\textrm{rel}\,\partial^{}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0\,,~X\!\not\succ_{\!}0_{}\}.$ + $\textrm{rel}\,\partial^{}\mathcal{K}=\{A\,{\text svec}X~|~X\!\succeq0\,,~X\!\not\succ_{\!}0_{}\}.$

Line 83: Line 83: Inverse image of the positive semidefinite cone under $\,g(y)\,$ Inverse image of the positive semidefinite cone under $\,g(y)\,$ - must therefore have dimension equal to $\dim\!\left(\mathcal{R}(A^{\rm T})_{}\!\cap\mbox{svec}\,\mathbb{S}_+^{_{}n}\right)$ + must therefore have dimension equal to $\dim\!\left(\mathcal{R}(A^{\rm T})_{}\!\cap{\text svec}\,\mathbb{S}_+^{_{}n}\right)$ and relative boundary and relative boundary - $\textrm{rel\,}\partial^{}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0\,,~\sum\limits_{j=1}^my_jA_j\not\succ_{}0_{}\}.$ + $\textrm{rel\,}\partial^{}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0\,,~\sum\limits_{j=1}^my_jA_j\not\succ0_{}\}.$ When this dimension is $\,m\,$, the dual cone interior is nonempty When this dimension is $\,m\,$, the dual cone interior is nonempty

Current revision

In convex optimization, a linear matrix inequality (LMI) is an expression of the form $LaTeX: LMI(y):=A_0+y_1A_1+y_2A_2+\ldots+y_m A_m\succeq0\,$

where

• $LaTeX: y=[y_i\,,~i\!=\!1\ldots m]$ is a real vector,
• $LaTeX: A_0\,, A_1\,, A_2\,,\ldots\,A_m$ 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.

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 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 $LaTeX: \,A_j\!\in\mathbb{S}^n$, $LaTeX: \,j\!=\!1\ldots m$ $LaTeX: \begin{array}{ll}\mathcal{K} \!\!&=\left\{\left[\begin{array}{c}\langle A_1\,,\,X^{}\rangle\\:\\\langle A_m\;,\,X^{}\rangle\end{array}\right]|~X\!\succeq_{\!}0\right\}\subseteq_{}\mathbb{R}^m\\\\ &=\left\{\left[\begin{array}{c}{\text svec}(A_1)^T\\:\\{\text svec}(A_m)^T\end{array}\right]{\text svec}X~|~X\!\succeq_{\!}0\right\}\\\\ &:=\;\{A\,{\text svec}X~|~X\!\succeq_{\!}0_{}\} \end{array}$

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 $LaTeX: A\,{\text svec}{X_{p_1}}_{\,},_{_{}}A\,{\text svec}{X_{p_2}}\!\in\mathcal{K}~\Rightarrow~ A(\zeta_{\,}{\text svec}{X_{p_1}}+_{}\xi_{\,}{\text svec}{X_{p_2}})\in_{}\mathcal{K} {\text~~for\,all~\,}\zeta_{\,},\xi\geq0$

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

Now consider the (closed convex) dual cone: $LaTeX: \begin{array}{rl}\mathcal{K}^* \!\!\!&=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z\!\in_{_{}\!}\mathcal{K}_{}\right\}\subseteq_{}\mathbb{R}^m\\ &=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z_{\!}=_{\!}A\,{\text svec}X\,,~X\succeq0_{}\right\}\\ &=_{}\left\{_{}y~|~\langle A\,{\text svec}X\,,~y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,X\!\succeq_{_{}\!}0_{}\right\}\\ &=\left\{y~|~\langle{\text svec}X\,,\,A^{T\!}y\rangle\geq_{}0\;~\textrm{for\,all}~\,X\!\succeq_{\!}0\right\}\\ &=\left\{y~|~{\text svec}^{-1}(A^{T\!}y)\succeq_{}0\right\} \end{array}$

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 $LaTeX: \,X\!\in\mathbb{S}_+^n\,.$

Relative interior of $LaTeX: \mathcal{K}$ may always be expressed $LaTeX: \textrm{rel\,int}\,\mathcal{K}=\{A\,{\text svec}X~|~X\!\succ0_{}\}.$

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

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

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

That is sufficient for convex cone $LaTeX: \,\mathcal{K}\,$ to be closed, and necessary to have relative boundary $LaTeX: \textrm{rel}\,\partial^{}\mathcal{K}=\{A\,{\text svec}X~|~X\!\succeq0\,,~X\!\not\succ_{\!}0_{}\}.$

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

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

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

and relative boundary $LaTeX: \textrm{rel\,}\partial^{}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0\,,~\sum\limits_{j=1}^my_jA_j\not\succ0_{}\}.$

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

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

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.