Linear matrix inequality

(Difference between revisions)
 Revision as of 15:50, 22 February 2009 (edit) (→Applications)← Previous diff Revision as of 19:27, 1 March 2009 (edit) (undo)Next diff → 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 cone defined over the positive semidefinite cone + Consider a peculiar vertex-description for a closed [[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 28: Line 28: *$A_0=\mathbf{0}$ is assumed without loss of generality. *$A_0=\mathbf{0}$ is assumed without loss of generality. - $\mathcal{K}$ is a convex cone because + $\mathcal{K}$ is a [[Convex cones|convex cone]] because $A\,\textrm{svec}{X_{{\rm p}_1}}_{\,},_{_{}}A\,\textrm{svec}{X_{{\rm p}_2}}\!\in\mathcal{K}~\Rightarrow~ [itex]A\,\textrm{svec}{X_{{\rm p}_1}}_{\,},_{_{}}A\,\textrm{svec}{X_{{\rm p}_2}}\!\in\mathcal{K}~\Rightarrow~ Line 71: Line 71: [itex]\,A\,\textrm{svec}X\,$ is an isomorphism in $\,X\,$ between the positive semidefinite cone $\mathbb{S}_+^n$ and range $\,\mathcal{R}(A)\,$ of matrix $\,A$. $\,A\,\textrm{svec}X\,$ 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 cone $\,\mathcal{K}\,$ has relative interior + In that case, [[Convex cones|convex cone]] $\,\mathcal{K}\,$ has relative interior $\textrm{rel\,int}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succ_{\!}0_{}\}$ $\textrm{rel\,int}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succ_{\!}0_{}\}$ Line 94: Line 94: == Applications == == 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. 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. + 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|convex cones]] governing this LMI. == External links == == External links ==

Revision as of 19:27, 1 March 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.

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.