# Questions tagged [linear-matrix-inequality]

Linear Matrix Inequalities (LMIs)

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• 123
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### Show that $\{x \mid A_0 + \sum x_i A_i \succcurlyeq 0 \}$ is convex

Let $A_0, A_1,\dots,A_m$ be symmetric matrices. Let $x \in \mathbb R^m$ and define $$A(x) := A_0 + \sum_{i=1}^m x_i A_i$$ Show that the set $C := \{x \mid A(x) \text{ is positive semidefinite} \}$ ...
• 1,440
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### Does the "Riccati LMI" implies boundedness?

Assume a matrix $P\succeq0$ satisfies the "Riccati LMI" \begin{align} \begin{pmatrix} FPF^T - P + M & FPH^T + S\\ HPF^T+S^T& HPH^T + I \end{pmatrix}\succeq0 \end{align} with $M\succ0$...
• 544
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### Trace minimization with additional constraint on $X$

Consider the following SDP problem: \begin{align*} \min_X \; & \mathrm{Tr}[AX]\\ \mathrm{s.t.}\; & X \succeq 0\\ & X \succeq \begin{bmatrix}0 & 0.5\\0.5 & 0\end{bmatrix}, \end{...
• 149
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### What is the meaning of the ≼ Symbol in the Context of Matrix Inequality with Symmetric Matrices

I saw the symbol ≼ in a textbook, and I am not quite sure what it means. The textbook says it represents matrix inequality, but again, I do not understand what that means. My best guesses are that the ...
• 101
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### How can you convert the following form of the Riccati equation into an LMI?

On page 3 of the EE363 notes on linear matrix inequalities, the following Riccati inequality is listed $$0 \leq A^TPA+Q-P-A^TPB(R+B^PB)^{-1}B^TPA, \qquad P \geq 0$$ I know that the Schur's ...
• 25
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### Common factor in sum of transpose matrices

I have the following linear matrix inequality: $(B K_1)^T + BK_1 < -2A$ where B is 2x1, $K_1$ is 1x2 and A is 2x2. Is it possible to find $K_1$ as a common factor in the left-hand side of the ...
1 vote
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### H2-filtering of unstable LTI systems. Can this problem be reformulated as convex optimization problem with LMI constraints?

Consider the discrete-time generalized LTI plant with minimal state-space realization $$x_{k+1}=A_d x_k + B_{d1}w_k\\z_k=C_{d1}x_k+D_{d11}w_k\\y_k=C_{d2}x_k+D_{d21}w_k$$ For the Schur-stable $A_d$ ...
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### LMI-Solution Invariant to the Initial Conditions

It is known from R. Bellman that the value of the functional $J = \int_{0}^{\infty}xWx \ dt, \ W>0$ along the solution of the linear time-invariant system $\dot x = Ax, \ x(0)=x_0$, with Schur ...
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• 25
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### How to check the feasibility of standard LMI using Matlab/CVX?

In the wikipedia page of LMI, the standard form is given by $$A_0+y_1A_1+y_2A_2+\cdots+y_mA_m \succeq 0,$$ where $A_i$ are $m\times m$ symmetric matrices and $y_i$ are real vectors, $i=1,2,\ldots m.$ ...
• 1,559
1 vote
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### Why is the boundary of spectrahedra “more pointy” at matrices of lower rank?

In the following expository article about spectrahedra, it is established informally that the boundary of spectrahedra is “more pointy” at matrices of lower rank. Cynthia Vinzant, What is a... ...
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### If $0\leq X \leq \text{Id}$ and $0\leq A$, then $XAX \leq A$?

Let $X,A\in\mathbb{C}^{n\times n}$ and suppose $$0 \leq X \leq \text{Id}, \quad 0\leq A,$$ where $\text{Id}\in\mathbb{C}^{n\times n}$ denotes the identity matrix. Is it true that $$XAX \leq A,$$ ...
• 650
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### Schur complement for linear matrix inequality (LMI)

Given the following inequality \begin{align} & \gamma \left( Q - (A Q + BY)^T Q^{-1} (A Q + BY) \right) - Y^T R^{1 \over 2} R^{1\over 2} Y - Q Q_1^{1\over2} Q_1^{1\over2} Q \succeq 0 \tag{1} \end{...
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### Mistake in Paper about LMI characterisation of trigonometric polynomial curve?

The following is taken from Efficient Large-Scale Filter/Filterbank Design via LMI Characterization of Trigonometric Curves by Hoang Duong Tuan, Tran Thai Son, Ba-Ngu Vo, and Truong Q. Nguyen Consider ...
• 4,214
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### A puzzling KKT for LMI vs. scalar constraint

I am trying to understand the KKT conditions for LMI constraints in order to solve my original question in KKT conditions for $\max \log \det(X)$ with LMI constraints. In the meantime, I found a much ...
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1 vote
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• 544
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• 585
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### Linear matrix inequality derivation from Risk-averse MPC problem

TLDR I need to use what looks like the Schur complement to transform a linear matrix inequality but instead of a $2\times 2$ block matrix there are more blocks. Question I'm having trouble with a ...
• 119
1 vote
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### Upper bounding the largest singular value of a matrix $X$ via LMI — is it correct?

For $z \in \Bbb C$ and $\delta > 0$, the inequality $|z| < \delta$ is equivalent to the matrix inequality $$\begin{bmatrix} -\delta & z\\ z^* & -\delta \end{bmatrix} \prec 0$$ (Source: ...
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### Why is the condition $\|Z\| < 1$ equivalent to $I - ZZ^{\top} > 0$?
As the title says, for a matrix $Z \in \mathbb{R}^{p \times q}$, the condition $\begin{Vmatrix}Z\end{Vmatrix} < 1$ equivalent to $I - ZZ^{\top} > 0$. How can I show the equivalence? Attempt: \$\...