Questions tagged [linear-matrix-inequality]

Linear Matrix Inequalities (LMIs)

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Constrained matrix equation

I have a data matrix $X \in \mathbb{R}^{n \times m}$, with $\mathrm{rank}(X) = n$ and $n \leq m$. I'm trying to understand if I can find some square matrix $C \in \mathbb{R}^{m \times m}$ such that $$ ...
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+50

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 ...
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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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1answer
30 views

Reformulation of LMI

In a paper I have read, the authors reformulated the following LMI $$X_{t} \succeq \left[\begin{array}{cc}\alpha_{i}\left(B_{t} U+C_{t}\right)^{\top} E_{i}\left(B_{t} U+C_{t}\right) & \left( B_{t} ...
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KKT conditions for $\max \log \det(X)$ with LMI constraints

I am trying to derive the KKT conditions for the following convex optimization problem where $A$ is a given matrix: $$\begin{array}{ll} \underset{X,Y,Z}{\text{minimize}} & - \log \det \left(I + Z +...
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1answer
27 views

The equivalence of a 2 by 2 positive semidefinite matrix and a 3 by 3 positive semidefinite matrix?

I came across the following: $$\begin{bmatrix} -x^TAx-2b^Tx+c &-(Ax+b)^TR\\ -R(Ax+b) & \lambda I -RAR \end{bmatrix}\geq0 \iff \begin{bmatrix} b^TA^{-1}b+c & 0 &(x+A^{-1}b)^T\\ 0 & ...
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1answer
22 views

Linear matrix inequality given the bounds

If $A\succeq A_{\min}\succ 0$, $B\succeq B_{\min}\succ 0$, will the following be true? $$ABA^\top \succeq AB_{\min}A^\top$$ How can I prove it?
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A problem combining linear and nonlinear matrix inequalities

I have a matrix inequality problem (don't know if this holds): $A \in \mathbb{R}^{m \times n}$, $B_i \in \mathbb{R}^{n \times n}$, $i=1, 2, \dots, m$. $B_i$ is Hermitian. $Ax>0$ doesn't have a ...
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1answer
46 views

Is there an efficient algorithm to determine if a linear matrix inequality has a solution?

Are there any fast algorithms to determine if a linear matrix inequality (LMI) problem $Ax \leq b$ has a solution? I am aware that linear programming and the simplex algorithm in particular may be ...
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equivalency in LMI formulation of LQR problem

In the LMI formulation of LQR problem, solving the following optimization problem is said to be equivalent to solving the original LQR problem: min trace(P) sbj to P>0 [A'PA+Q−P A'PB; B'PA R+B'PB]⪰...
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Can $X - Y A^\dagger Y^T\succ0$ be written as an LMI where $A^\dagger$ is a pseudoinverse?

I have the constraint \begin{align} X - Y A^\dagger Y^T\succ0, \end{align} where $A^\dagger$ is the pseudoinverse of $A\succeq0$. Can we still use the Schur complement to write the constraint as an ...
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54 views

If $A\succ0$ and $J$ is the matrix of ones, then $A \succeq J$ if and only if $\mathrm{trace}(A^{-1}J)\leq 1$

I'm trying to prove that if $A \in \mathbb{R}^{n \times n}$ is positive definite, and $J$ is the $n \times n$ matrix of ones, then $A \succeq J$ if and only if $\mathrm{trace}(A^{-1}J) = \sum_{i,j} A^{...
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1answer
18 views

Is the direction of inequality preserved when multiplying the linear matrix inequality (LMI) by a nonsingular square matrix left and right?

Let $Y$ be a square and nonsingular matrix (invertible). Given $X \prec 0$, if we multiply $Y^*$ on the left and $Y$ on the right to the $X$ (assuming the dimension matches), I have a claim that $$ X \...
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37 views

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 ...
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35 views

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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2answers
51 views

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: $\...
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20 views

Does linear matrix inequality imply eigenvalue inequality? [duplicate]

Suppose $A, B\in S^n$ and $A\succeq B$. Further suppose the eigenvalues of $A$ in non-increasing order are $\lambda_1, \lambda_2, ..., \lambda_n$, and $B$'s are $\mu_1, ..., \mu_n$. Does it imply $\...
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1answer
47 views

Matrix operator norm inequality proof

For any matrix $A \in \mathbb{R}^{m,n}$ and any $u \in \mathbb{R}^{m}$ and $v \in \mathbb{R}^n$, how to proof the following inequality? $$ |u^\top A v | \leq \|A\|_{op} \|u\|_1 \|v\|_1 $$ In general, ...
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1answer
46 views

Sufficient Conditions $DA^T+AD+I<0$ to Hold Based on Eigenvalues

I want to find sufficient conditions for the following matrix inequality to hold: $$DA^T+AD+I<0$$ based on the eigenvalues of $A$ and $D$, where $D$ is diagonal with negative entries and $A$ is ...
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1answer
89 views

matrix inner product between positive semidefinite matrix and positive definite matrix

Let $F_0, F_1, \ldots, F_m$ be a $n \times n$ symmetric matrices. We define $$F(x) := F_0 + x_1 F_1 + \cdots + x_m F_m$$ Show that if there does not exist $x \in \Bbb R^m$ such that $F(x)$ is positive ...
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1answer
37 views

Inequality for the trace of the n-th power of a semi-definite matrix with trace smaller than 1.

Let $M$ be a $mxm$ positive semi-definite matrix with $tr(M)\le1$. Is there some non-trivial inequality of the type $tr(M^n)\ge f(tr(M^i), tr(M^j))$ with $f$ some function and $i,j\le n$ ? And what ...
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32 views

Formulation of an Optimization Problem

I have the following optimization problem: \begin{equation} \label{lip1} \begin{aligned} \max \lambda \ \ \ \ \text{s.t.} \\ \begin{bmatrix} (AX+BY)^T+AX+BY+\lambda_0 X & B_w\\ * & -\mu ...
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Transformations in LMI constraints

Suppose that $T$ is a symmetric and invertible matrix. The following must be true \begin{equation}\label{eq:SDP}\left[\begin{array}{cc} X & A \\ A^{\top} & B \end{array}\right] \succeq 0 \...
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29 views

Linear matrix inequality (LMI) and Schur complement

I am trying to simplify the matrix especially the element in 1by1 position. (the blue circle in the image and the * is transposed term such as [A B|B^T C]=[A B|* C]). Simplifying means that I want to ...
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1answer
178 views

Spectral norm - trace inequality

I am wondering whether the following is true under which assumptions on A and B? $\operatorname{trace}(AB)\leqslant\|A\| \operatorname{trace}(B)$ The matrix norm is the spectral norm here. Maybe ...
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Reformulation to LMI constraints (H2 Feedforward Control)

I am trying to solve a problem similar to a H2 feedforward control synthesis problem with an additional constraint. I was wondering if it is possible to reformulate the problem to be a set of linear ...
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3answers
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Relation between two linear matrix inequality (LMI) problems

The following $>, \geq$ signs are referred to positive definiteness. Given an arbitrary matrix $A\in \mathbb{R}^{n\times n} $. Problem 1: Find a matrix $P=P^\text{T}\in\mathbb{R}^{n\times n}$, $P&...
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1answer
47 views

Inequality to find the condition for positive semi-definite

I have a question regarding the convexity of the system. The formulation I arrived at has the Hessian of the following form, $$A^TKA = \begin{bmatrix}x_1 & x_2 & ... & x_n \\ y_1 & ...
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1answer
43 views

A matrix inequality between two positive definite matrices and a positive scalar.

I am trying to prove the following inequality: $$ P R^{-1} P - 2 \epsilon P + \epsilon^2 R \geq 0, $$ where matrices $P$ and $R$ are definite positive, and $\epsilon$ is a real-valued positive ...
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1answer
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Reference to this Young Inequality for matrices

In this post: here I saw the usage of this Young Inequality: $$X^TY+Y^TX\leq \frac{1}{2}(X+SY)^TS^{-1}(X+SY)$$ With S any symmetric positive definite matrix. As far as I understood this could help ...
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Converting nonlinear matrix inequality to an LMI

I am fairly new to semidefinite programming (SDP). I have the following semidefinite program in matrix $X$ and vector $p$ $$\begin{array}{ll} \text{minimize} & \|Np\|^2\\ \text{subject to} & ...
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Variant of the Schur Complement

Let $$M = \left(\begin{array}{cc}{A} & {B} \\ {B^{T}} & {C}\end{array}\right)$$ be symmetric, and let $A$ be invertible. Then the Schur Complement Lemma suggests that $$C-B^{T} A^{-1} B \...
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Algorithm for LMI optimization of PSD matrix

To optimize a PSD $(n,n)$ matrix $ X $, we are asked us to propose (give ), implement and compare different Algorithms for the resolution of a Linear matrix inequality given by: $$\begin{array}{ll} \...
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1answer
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Schur complements for nonstrict inequalities

I am trying to understand the following proof from the book "Linear Matrix Inequalities in System and Control Theory". However I am struggling to understand why $S_{2}$ must equal zero. Why isn'...
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1answer
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LMI and singular value [duplicate]

Can we have the following? For matrices A and B, if A⪰B⟹σ¯¯¯(A)≥σ¯¯¯(B) ? where σ¯¯¯(⋅) means the largest singular value.
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1answer
47 views

Singular value relation for an LMI

Can we have the following? For matrices $A$ and $B$, if $A \succeq B \implies \overline\sigma(A) \ge \overline\sigma(B)$? where $\overline\sigma(\cdot)$ means the largest singular value.
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1answer
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How to rewrite the nonlinear matrix inequality as a linear one?

Let $t>0$ be a scalar, $P \in \mathbb{R}^{n\times n}$ is a symmetric positive-definite matrix, and $B \in \mathbb{R}^{n\times m}$ for some integers $n$ and $m$. I have the matrix inequality $tP^2-B ...
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Formulating a convex optimization problem as semidefinite program

I have the following minimization problem $$\text{minimize} \quad f(x)= c^T F(x)^{-1} c$$ where $F : \mathbb R^n \to \mbox{Sym}_m (\mathbb R)$, $$\mbox{dom} f = \{x \in \mathbb{R}^n \mid F(x) \succ ...
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How can we write this inequality as an LMI?

Let $X$, $Y$ and $Z$ be positive definite matrices. How can we write the following inequality as an LMI? $$XY - Z^2 - I \succ 0$$ Here, $I$ is the identity matrix. For example, if it was $XY-Z^2Y-I&...
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1answer
275 views

Optimization with matrix inverse constraint

I have a MILP problem. Now I want to include a new constraint to it (upper bound the diagonal elements of an inversed matrix), i.e., ${Y(x)^{-1}}_{ii}<k_i, \forall i\in\{1,2,...,n\}$ where $Y(x)\...
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1answer
231 views

Minimizing trace of matrix inverse is an SDP?

I'm reading through this paper and on page 8 (PDF page 9), Step 2 in Algorithm 1 reads $$ W \leftarrow \text{argmin} \{\text{tr}(WQW^T) + \alpha\, \text{tr}(Q^{−1}): Q \succeq \epsilon\, I_n , Q_{\...
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1answer
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About the use of Schur's complement. Why they are equivalent?

$$y^{\top}Qy+y^{\top}q+r\geq -ay^{\top}x-b, \qquad \forall y \in \mathbb R^{n}$$ where $Q \succeq 0$. One can use Schur's complement to replace it by an equivalent linear matrix inequality (LMI). $$...
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LMI-based versus standard form semidefinite programs

In the context of semidefinite programming (SDP), under what conditions is it preferable to formulate and solve an LMI-based SDP rather than an equivalent standard form SDP? I have been told that ...
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1answer
160 views

using Schur's complement and Young's inequality to reduce matrix algebraic equation to LMI

This questions concerns the practical implementation of schur's complement and Young's inequality. Consider the following \begin{align} \begin{pmatrix} \begin{pmatrix} \mathbb{A}_{\mathbb{Z}} & ...
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2answers
68 views

Finite horizon Riccati solved by LMI

Considering the system $x_{k+1}=Ax_k+Bu_k$ with quadratic cost $J^* = \min x_N^T S x_N + \sum_{k=0}^{N-1} x_k^T Qx_k+u^T_kRu_k$ where $Q,S\succeq 0, R\succ 0$. The optimal state feedback is found ...
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2answers
85 views

Schur complement without inverted term and YALMIP solving

first of all, excuse my poor English :( Anyway, I need to factorize the equation below in terms of Schur Complement. $\begin{equation} { x }^{ T }\left( PA+{ A }^{ T }P \right) x<-\gamma { x }^{ ...
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1answer
61 views

Simplest way to prove that the following constraint is convex [closed]

We have symmetric and positive semidefinite matrix variables $W_1, W_2, \dots, W_K$. Further, we have $$Q_i = W_i - \sum_{j=1,j \neq i}^K W_j$$ and a constant real vector $x$. How to show that the ...
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54 views

Convert Discrete Riccati Equation to LMI for multi-model LQG problem

I am interested in solving the multi-model discrete-time LQG problem, that is to synthesize a stabilizing output feedback controller that stabilizes all systems with matrices $A_i, B_i,C_i,D_i$ with $...
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1answer
279 views

Maximize the smallest nonzero singular value

I want to maximize the smallest nonzero singular value of (non-square) matrix $X$. This is equivalent to maximizing $\lambda_{\min}(X^\top X)$, which can be reformulated as follows $$\begin{array}{ll}...
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1answer
140 views

Convert the following Riccati equation into an LMI

I need help to convert the following discrete-time Riccati eqn. into an LMI with two decision variables $(K, P)$: $$ (A+BK)^T P (A+BK) - P +Q + K^T R K \prec 0$$ The trick, at least for the ...