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 - * $y=[y_i\,,~i\!=\!1\dots m]$ is a real vector, + * $y=[y_i\,,~i\!=\!1\ldots m]$ 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 $\mathbb{S}^n$, + * $A_0\,, A_1\,, A_2\,,\ldots\,A_m$ 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 $\mathbb{S}$. * $B\succeq0$ 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 $\mathbb{S}$. Line 11: Line 11: $LMI(y)\succeq 0$ 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]''')''' $LMI(y)\succeq 0$ 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 cones|convex cone]] defined over the positive semidefinite cone + Consider a peculiar vertex-description for a [[Convex cones|convex cone]] defined over the positive semidefinite cone '''('''instead of the more common nonnegative orthant, $x\succeq0$''')''': '''('''instead of the more common nonnegative orthant, $x\succeq0$''')''': Line 18: Line 18: $\begin{array}{ll}\mathcal{K} [itex]\begin{array}{ll}\mathcal{K} - \!\!&=\left\{\left[\begin{array}{c}\langle A_1\,,\,X^{}\rangle\\\vdots\\\langle A_m\;,\,X^{}\rangle\end{array}\right]|~X\!\succeq_{\!}0\right\}\subseteq_{}\reals^m\\\\ + \!\!&=\left\{\left[\begin{array}{c}\langle A_1\,,\,X^{}\rangle\\:\\\langle A_m\;,\,X^{}\rangle\end{array}\right]|~X\!\succeq_{\!}0\right\}\subseteq_{}\mathbb{R}^m\\\\ - &=\left\{\left[\begin{array}{c}\textrm{svec}(A_1)^T\\\vdots\\\textrm{svec}(A_m)^T\end{array}\right]\!\textrm{svec}X~|~X\!\succeq_{\!}0\right\}\\\\ + &=\left\{\left[\begin{array}{c}{\text svec}(A_1)^T\\:\\{\text svec}(A_m)^T\end{array}\right]{\text svec}X~|~X\!\succeq_{\!}0\right\}\\\\ - &:=\;\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0_{}\} + &:=\;\{A\,{\text svec}X~|~X\!\succeq_{\!}0_{}\} \end{array}$ \end{array}[/itex] where where *$A\!\in_{}\!\mathbb{R}^{m\times n(n+1)/2}$, *$A\!\in_{}\!\mathbb{R}^{m\times n(n+1)/2}$, - *symmetric vectorization svec is a stacking of columns defined in '''('''[http://meboo.convexoptimization.com/Meboo.html Dattorro, Ch.2.2.2.1]''')''', + *symmetric vectorization svec is a stacking of columns defined in '''('''[http://meboo.convexoptimization.com/Meboo.html Dattorro, ch.2.2.2.1]''')''', *$A_0=\mathbf{0}$ is assumed without loss of generality. *$A_0=\mathbf{0}$ is assumed without loss of generality. $\mathcal{K}$ 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 39: Line 39: $\begin{array}{rl}\mathcal{K}^* [itex]\begin{array}{rl}\mathcal{K}^* - \!\!\!&=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z\!\in_{_{}\!}\mathcal{K}_{}\right\}\subseteq_{}\reals^m\\ + \!\!\!&=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z\!\in_{_{}\!}\mathcal{K}_{}\right\}\subseteq_{}\mathbb{R}^m\\ - &=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z_{\!}=_{\!}A\,\textrm{svec}X\,,~X\succeq0_{}\right\}\\ + &=_{}\left\{_{}y~|~\langle z\,,\,y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,z_{\!}=_{\!}A\,{\text svec}X\,,~X\succeq0_{}\right\}\\ - &=_{}\left\{_{}y~|~\langle A\,\textrm{svec}X\,,~y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,X\!\succeq_{_{}\!}0_{}\right\}\\ + &=_{}\left\{_{}y~|~\langle A\,{\text svec}X\,,~y_{}\rangle\geq_{}0\,~\textrm{for\,all}~\,X\!\succeq_{_{}\!}0_{}\right\}\\ - &=\left\{y~|~\langle\textrm{svec}X\,,\,A^{T\!}y\rangle\geq_{}0\;~\textrm{for\,all}~\,X\!\succeq_{\!}0\right\}\\ + &=\left\{y~|~\langle{\text svec}X\,,\,A^{T\!}y\rangle\geq_{}0\;~\textrm{for\,all}~\,X\!\succeq_{\!}0\right\}\\ - &=\left\{y~|~\textrm{svec}^{-1}(A^{T\!}y)\succeq_{}0\right\} + &=\left\{y~|~{\text svec}^{-1}(A^{T\!}y)\succeq_{}0\right\} \end{array}$ \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 60: Line 60: Although matrix $\,A\,$ is finite-dimensional, $\mathcal{K}$ is generally not a polyhedral cone Although matrix $\,A\,$ is finite-dimensional, $\mathcal{K}$ is generally not a polyhedral cone - (unless $\,m\,$ equals 1 or 2) simply because $\,X\!\in\mathbb{S}_+^n\,$. + (unless $\,m\,$ equals 1 or 2) simply because $\,X\!\in\mathbb{S}_+^n\,.$ - Provided the $A_j$ matrices are linearly independent, then relative interior = interior + Relative interior of $\mathcal{K}$ may always be expressed + $\textrm{rel\,int}\,\mathcal{K}=\{A\,{\text svec}X~|~X\!\succ0_{}\}.$ - $\textrm{rel\,int}\mathcal{K}=\textrm{int}\mathcal{K}$ + Provided the $\,A_j$ matrices are linearly independent, then + $\textrm{rel\,int}\,\mathcal{K}=\textrm{int}\,\mathcal{K}$ - meaning, the cone interior is nonempty; implying, the dual cone is pointed ([http://meboo.convexoptimization.com/Meboo.html Dattorro, ch.2]). + meaning, cone $\mathcal{K}$ interior is nonempty; implying, dual cone $\mathcal{K}^*$ is pointed ([http://meboo.convexoptimization.com/Meboo.html Dattorro, ch.2]). - If matrix $\,A\,$ has no nullspace, on the other hand, 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 $\,A.$ - In that case, [[Convex cones|convex cone]] $\,\mathcal{K}\,$ has relative interior + That is sufficient for [[Convex cones|convex cone]] $\,\mathcal{K}\,$ to be closed, and necessary to have relative boundary + $\textrm{rel}\,\partial^{}\mathcal{K}=\{A\,{\text svec}X~|~X\!\succeq0\,,~X\!\not\succ_{\!}0_{}\}.$ - $\textrm{rel\,int}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succ_{\!}0_{}\}$ +
+ Relative interior of the dual cone may always be expressed + $\textrm{rel\,int}\,\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succ_{}0_{}\}.$ - and boundary + When the $A_j$ matrices are linearly independent, function $\,g(y)_{\!}:=_{_{}\!}\sum y_jA_j\,$ is a linear bijection on $\mathbb{R}^m.$ - + - $\textrm{rel}\,\partial^{}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0\,,~X\!\nsucc_{\!}0_{}\}$ + - + -
When the $A_j$ matrices are linearly independent, function $\,g(y)_{\!}:=_{_{}\!}\sum y_jA_j\,$ on $\mathbb{R}^m$ is a linear bijection. + Inverse image of the positive semidefinite cone under $\,g(y)\,$ Inverse image of the positive semidefinite cone under $\,g(y)\,$ - must therefore have dimension $_{}m$. + must therefore have dimension equal to $\dim\!\left(\mathcal{R}(A^{\rm T})_{}\!\cap{\text svec}\,\mathbb{S}_+^{_{}n}\right)$ - + - In that circumstance, the dual cone interior is nonempty + - $\textrm{int}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succ_{}0_{}\}$ + and relative boundary + $\textrm{rel\,}\partial^{}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0\,,~\sum\limits_{j=1}^my_jA_j\not\succ0_{}\}.$ - having boundary + When this dimension is $\,m\,$, the dual cone interior is nonempty + $\textrm{rel\,int}\,\mathcal{K}^*=\textrm{int}\,\mathcal{K}^*$ - $\partial^{}\mathcal{K}^*=\{y\!\in_{}\!\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0\,,~\sum\limits_{j=1}^my_jA_j\nsucc_{}0_{}\}$ + and closure of convex cone $\mathcal{K}$ is pointed. == Applications == == Applications == Line 97: Line 98: == External links == == External links == - * 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] - * 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). + * C. Scherer and S. Weiland, [http://w3.ele.tue.nl/nl/cs/education/courses/hyconlmi Course on Linear Matrix Inequalities in Control], Dutch Institute of Systems and Control (DISC).

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.