Linear matrix inequality - Revision history

Ranjelin: /* Convexity of the LMI constraint */

2026-05-09T22:56:28Z

Convexity of the LMI constraint

←Older revision		Revision as of 22:56, 9 May 2026
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	This leads directly to an equally peculiar halfspace-description		This leads directly to an equally peculiar halfspace-description

-	<math>\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \sum y_j A_j \succeq 0\}</math>	+	<math>\mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \sum y_j A_j \succeq 0\}</math>

	The summation inequality with respect to the positive semidefinite cone		The summation inequality with respect to the positive semidefinite cone

Ranjelin: /* LMI Geometry */

2026-05-09T22:54:48Z

LMI Geometry

←Older revision		Revision as of 22:54, 9 May 2026
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	Relative interior of <math>\mathcal{K}</math> may always be expressed		Relative interior of <math>\mathcal{K}</math> may always be expressed
-	<math>{\rm rel\,int}\,\mathcal{K}=\{A\,{\rm svec}X\;\|\;X\~~!\succ_{\!}0_{}~~\}</math>.	+	<math>{\rm rel\,int}\,\mathcal{K}=\{A\,{\rm svec}X\;\|\;X\succ0\}</math>.

	Provided the <math>\,A_j</math> matrices are linearly independent, then		Provided the <math>\,A_j</math> matrices are linearly independent, then
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	That is sufficient for [[Convex cones\|convex cone]] <math>\,\mathcal{K}\,</math> to be closed, and necessary to have relative boundary		That is sufficient for [[Convex cones\|convex cone]] <math>\,\mathcal{K}\,</math> to be closed, and necessary to have relative boundary
-	<math>{\rm rel}\,\partial^{}\mathcal{K}=\{A\,{\rm svec}X\;\|\;X\~~!\succeq_{\!}0~~ \,,\; X \not \succ ~~0_{}~~\}</math>.	+	<math>{\rm rel}\,\partial^{}\mathcal{K}=\{A\,{\rm svec}X\;\|\;X\succeq0 \,,\; X \not \succ 0\}</math>.

	<br>		<br>

Ranjelin: /* LMI Geometry */

2026-05-06T10:02:57Z

LMI Geometry

←Older revision		Revision as of 10:02, 6 May 2026
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	<br>		<br>
	Relative interior of the dual cone may always be expressed		Relative interior of the dual cone may always be expressed
-	<math>{\rm rel\,int}\,\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \~~sum_j~~ y_j A_j \succ 0\}</math>.	+	<math>{\rm rel\,int}\,\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \sum y_j A_j \succ 0\}</math>.

	When the <math>A_j</math> matrices are linearly independent, function <math>\,g(y)\,:=\,\sum y_j A_j\,</math> is a linear bijection on <math>{\bb{R}}^m</math>.		When the <math>A_j</math> matrices are linearly independent, function <math>\,g(y)\,:=\,\sum y_j A_j\,</math> is a linear bijection on <math>{\bb{R}}^m</math>.
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	and relative boundary		and relative boundary
-	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \~~sum_j~~ y_j A_j \succeq 0 \,,\, \~~sum_j~~ y_j A_j \not \succ 0\}</math>.	+	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \sum y_j A_j \succeq 0 \,,\, \sum y_j A_j \not \succ 0\}</math>.

	When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty		When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty

Ranjelin: /* Convexity of the LMI constraint */

2026-05-06T10:01:43Z

Convexity of the LMI constraint

←Older revision		Revision as of 10:01, 6 May 2026
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	This leads directly to an equally peculiar halfspace-description		This leads directly to an equally peculiar halfspace-description

-	<math>\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \~~sum_j~~ y_j A_j \succeq 0\}</math>	+	<math>\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \sum y_j A_j \succeq 0\}</math>

	The summation inequality with respect to the positive semidefinite cone		The summation inequality with respect to the positive semidefinite cone

Ranjelin at 02:18, 6 May 2026

2026-05-06T02:18:19Z

←Older revision		Revision as of 02:18, 6 May 2026
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	: <math>LMI(y)\,:=A_0 \,+\, y_1 A_1 \,+\, y_2 A_2 \,+\, \ldots \,+\, y_m A_m\succeq0</math>		: <math>LMI(y)\,:=A_0 \,+\, y_1 A_1 \,+\, y_2 A_2 \,+\, \ldots \,+\, y_m A_m\succeq0</math>
	where		where
-	* <math>y=[~~y_i~~\,,\,i\!=1\,\ldots\,m]</math> is a real vector,	+	* <math>y=[y_j\,,\,j\!=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>{\bb{S}}^n</math>,		* <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>{\bb{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>{\bb{S}}_+</math> in the subspace of symmetric matrices <math>{\bb{S}}</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>{\bb{S}}_+</math> in the subspace of symmetric matrices <math>{\bb{S}}</math>.

Ranjelin: /* Convexity of the LMI constraint */

2026-05-06T02:15:21Z

Convexity of the LMI constraint

←Older revision		Revision as of 02:15, 6 May 2026
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	This leads directly to an equally peculiar halfspace-description		This leads directly to an equally peculiar halfspace-description

-	<math>\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \~~sum _{j= 1} ^m~~ y_j A_j \succeq 0\}</math>	+	<math>\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \sum_j y_j A_j \succeq 0\}</math>

	The summation inequality with respect to the positive semidefinite cone		The summation inequality with respect to the positive semidefinite cone

Ranjelin: /* LMI Geometry */

2026-05-06T02:14:34Z

LMI Geometry

←Older revision		Revision as of 02:14, 6 May 2026
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	<br>		<br>
	Relative interior of the dual cone may always be expressed		Relative interior of the dual cone may always be expressed
-	<math>{\rm rel\,int}\,\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \~~sum _{j = 1} ^m~~ y_j A_j \succ 0\}</math>.	+	<math>{\rm rel\,int}\,\mathcal{K}^* = \{y \in {\bb{R}}^m \;\|\, \sum_j y_j A_j \succ 0\}</math>.

	When the <math>A_j</math> matrices are linearly independent, function <math>\,g(y)\,:=\,\sum y_j A_j\,</math> is a linear bijection on <math>{\bb{R}}^m</math>.		When the <math>A_j</math> matrices are linearly independent, function <math>\,g(y)\,:=\,\sum y_j A_j\,</math> is a linear bijection on <math>{\bb{R}}^m</math>.
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	and relative boundary		and relative boundary
-	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \~~sum _{j ~=~ 1} ^m~~ y_j A_j \succeq 0 \,,\, \~~sum _{j = 1} ^m~~ y_j A_j \not \succ 0\}</math>.	+	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \sum_j y_j A_j \succeq 0 \,,\, \sum_j y_j A_j \not \succ 0\}</math>.

	When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty		When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty

Ranjelin: /* Convexity of the LMI constraint */

2026-05-06T00:09:04Z

Convexity of the LMI constraint

←Older revision		Revision as of 00:09, 6 May 2026
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	== Convexity of the LMI constraint ==		== 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]''')'''	+	<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 2.13.6.1.1]''')'''

	Consider a peculiar vertex-description for a [[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

Ranjelin: /* LMI Geometry */

2026-05-04T02:58:42Z

LMI Geometry

←Older revision		Revision as of 02:58, 4 May 2026
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	and relative boundary		and relative boundary
-	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \sum _{j~= 1} ^m y_j A_j \succeq 0 \,,\, \sum _{j = 1} ^m y_j A_j \not \succ 0\}</math>.	+	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \sum _{j ~=~ 1} ^m y_j A_j \succeq 0 \,,\, \sum _{j = 1} ^m y_j A_j \not \succ 0\}</math>.

	When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty		When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty

Ranjelin: /* LMI Geometry */

2026-05-04T01:17:47Z

LMI Geometry

←Older revision		Revision as of 01:17, 4 May 2026
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	and relative boundary		and relative boundary
-	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \sum _{j= 1} ^m y_j A_j \succeq 0 \,,\, \sum _{j = 1} ^m y_j A_j \not \succ 0\}</math>.	+	<math>{\rm rel\,} \partial \mathcal{K}^* = \{y \in {\bb{R}}^m \,\|\, \sum _{j~= 1} ^m y_j A_j \succeq 0 \,,\, \sum _{j = 1} ^m y_j A_j \not \succ 0\}</math>.

	When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty		When this dimension is <math>\,m\,</math>, the dual cone interior is nonempty