Ranjelin: Undo revision 3349 by Ranjelin (Talk)

Ranjelin — Sun, 19 Apr 2026 01:01:18 GMT

Undo revision 3349 by Ranjelin (Talk)

←Older revision		Revision as of 01:01, 19 April 2026
Line 17:		Line 17:
	for <math>X\!\in\mathbb{S}^n</math> given <math>\,A_j\!\in\mathbb{S}^n</math>, <math>\,j\!=\!1\ldots m</math>		for <math>X\!\in\mathbb{S}^n</math> given <math>\,A_j\!\in\mathbb{S}^n</math>, <math>\,j\!=\!1\ldots m</math>

-	<math>\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}{\rm svec}(A_1)^{\rm T}\\\vdots\\{\rm svec}(A_m)^{\rm T}\end{array}\right]{\rm svec}X~\|~X\succeq 0\right\}\\\\&:=\;\{A\,{\rm svec}X~\|~X\succeq 0\}\end{array}</math>	+	<math>\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}{\rm svec}(A_1)^{\rm T}\\\vdots\\{\rm svec}(A_m)^{\rm T}\end{array}\right]{\rm svec}X~\|~X\succeq 0\right\}\\\\
		+	&:=\;\{A\,{\rm svec}X~\|~X\succeq 0\}
		+	\end{array}</math>

	where		where

Ranjelin at 01:00, 19 April 2026

Ranjelin — Sun, 19 Apr 2026 01:00:26 GMT

←Older revision		Revision as of 01:00, 19 April 2026
Line 17:		Line 17:
	for <math>X\!\in\mathbb{S}^n</math> given <math>\,A_j\!\in\mathbb{S}^n</math>, <math>\,j\!=\!1\ldots m</math>		for <math>X\!\in\mathbb{S}^n</math> given <math>\,A_j\!\in\mathbb{S}^n</math>, <math>\,j\!=\!1\ldots m</math>

-	<math>\begin{array}{ll}\mathcal{K}	+	<math>\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}{\rm svec}(A_1)^{\rm T}\\\vdots\\{\rm svec}(A_m)^{\rm T}\end{array}\right]{\rm svec}X~\|~X\succeq 0\right\}\\\\&:=\;\{A\,{\rm svec}X~\|~X\succeq 0\}\end{array}</math>
-	&=\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}{\rm svec}(A_1)^{\rm T}\\\vdots\\{\rm svec}(A_m)^{\rm T}\end{array}\right]{\rm svec}X~\|~X\succeq 0\right\}\\\\	+
-	&:=\;\{A\,{\rm svec}X~\|~X\succeq 0\}	+
-	\end{array}</math>	+

	where		where

Ranjelin: New page: In convex optimization, a '''linear matrix inequality (LMI)''' is an expression of the form : $LMI(y):=A_0+y_1A_1+y_2A_2+\ldots+y_m A_m\succeq0\,$ where * $y=[y_i\,,~i\!=\!...$

Ranjelin — Sun, 15 Feb 2026 22:19:27 GMT

New page: In convex optimization, a '''linear matrix inequality (LMI)''' is an expression of the form : <math>LMI(y):=A_0+y_1A_1+y_2A_2+\ldots+y_m A_m\succeq0\,</math> where * <math>y=[y_i\,,~i\!=\!...

New page

In convex optimization, a '''linear matrix inequality (LMI)''' is an expression of the form
: <math>LMI(y):=A_0+y_1A_1+y_2A_2+\ldots+y_m A_m\succeq0\,</math>
where
* <math>y=[y_i\,,~i\!=\!1\ldots m]</math> is a real vector,
* <math>A_0\,, A_1\,, A_2\,,\ldots\,A_m</math> are symmetric matrices in the subspace of <math>n\times n</math> symmetric matrices <math>\mathbb{S}^n</math>,
* <math>B\succeq0 </math> is a generalized inequality meaning <math>B</math> is a positive semidefinite matrix belonging to the positive semidefinite cone <math>\mathbb{S}_+</math> in the subspace of symmetric matrices <math>\mathbb{S}</math>.

This linear matrix inequality specifies a convex constraint on ''y''.

== Convexity of the LMI constraint ==
<math>LMI(y)\succeq 0</math> 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 [[Convex cones|convex cone]] defined over the positive semidefinite cone

'''('''instead of the more common nonnegative orthant, <math>x\succeq0</math>''')''':

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

<math>\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}{\rm svec}(A_1)^{\rm T}\\\vdots\\{\rm svec}(A_m)^{\rm T}\end{array}\right]{\rm svec}X~|~X\succeq 0\right\}\\\\
&:=\;\{A\,{\rm svec}X~|~X\succeq 0\}
\end{array}</math>

where
*<math>A\!\in_{}\!\mathbb{R}^{m\times n(n+1)/2}</math>,
*symmetric vectorization svec is a stacking of columns defined in '''('''[http://meboo.convexoptimization.com/Meboo.html Dattorro, ch.2.2.2.1]''')''',
*<math>A_0=\mathbf{0}</math> is assumed without loss of generality.

<math>\mathcal{K}</math> is a [[Convex cones|convex cone]] because

<math>A\,\textrm{svec}{X_{_{p_1}}}_{\,},_{_{}}A\,\textrm{svec}{X_{_{p_2}}}\!\in\mathcal{K}~\Rightarrow~
A(\zeta_{\,}\textrm{svec}{X_{_{p_1}\!}}+_{}\xi_{\,}\textrm{svec}{X_{_{p_2}}})\in_{}\mathcal{K}
\textrm{~~for\,all~\,}\zeta_{\,},\xi\geq0</math>

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

Now consider the (closed convex) dual cone:

<math>\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\,\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}</math>

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

* <math>Y\succeq0~\Leftrightarrow~\langle Y\,,\,X\rangle\geq0\;~\textrm{for\,all}~\,X\succeq0</math>

This leads directly to an equally peculiar halfspace-description

<math>\mathcal{K}^*=\{y\!\in\mathbb{R}^m~|\,\sum\limits_{j=1}^my_jA_j\succeq_{}0_{}\}</math>

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

== LMI Geometry ==

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

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

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

meaning, cone <math>\mathcal{K}</math> interior is nonempty; implying, dual cone <math>\mathcal{K}^*</math> is pointed ([http://meboo.convexoptimization.com/Meboo.html Dattorro, ch.2]).

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

That is sufficient for [[Convex cones|convex cone]] <math>\,\mathcal{K}\,</math> to be closed, and necessary to have relative boundary
<math>\textrm{rel}\,\partial^{}\mathcal{K}=\{A\,\textrm{svec}X~|~X\!\succeq_{\!}0\,,~X\!\not\succ_{\!}0_{}\}.</math>

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

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

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

and relative boundary
<math>{\rm rel\,}\partial\mathcal{K}^*=\{y\in\mathbb{R}^m~|\,\sum\limits_{j=1}^m y_j A_j\succeq 0\,,~\sum\limits_{j=1}^m y_j A_j\not\succ 0\}</math>

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

and closure of convex cone <math>\mathcal{K}</math> is pointed.

== Applications ==
There are efficient numerical methods to determine whether an LMI is feasible (''i.e.'', whether there exists a vector <math>y</math> such that <math>LMI(y)\succeq0</math> ), 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|convex cones]] governing this LMI.

== 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]

* 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).

Linear Matrix Inequality II - Revision history

Ranjelin: Undo revision 3349 by Ranjelin (Talk)

Ranjelin at 01:00, 19 April 2026

Ranjelin: New page: In convex optimization, a '''linear matrix inequality (LMI)''' is an expression of the form : LMI(y):=A_0+y_1A_1+y_2A_2+\ldots+y_m A_m\succeq0\, where * y=[y_i\,,~i\!=\!...

Ranjelin: New page: In convex optimization, a '''linear matrix inequality (LMI)''' is an expression of the form : $LMI(y):=A_0+y_1A_1+y_2A_2+\ldots+y_m A_m\succeq0\,$ where * $y=[y_i\,,~i\!=\!...$