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Questions tagged [lmis]

Linear Matrix Inequalities (LMIs)

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26 views

Can this Matrix Inequality be convexified/linearized?

(see Picture) I have the following matrix inequality in \Pi, U, V=V^T, and W=W^T, where size(V)>size(W). I would like to know whether this MI can be linearized. Any suggestions are welcome... If any ...
0
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1answer
39 views

Describing Constraints Using Linear Algebra (Convex Optimization)

I've been learning Convex Optimization but one thing that really confused me in class was how exactly to recast a given set of constraints in matrix form, so that it can be solved using CVX. For ...
0
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1answer
35 views

How to write this inequality in terms of Schur Complement?

I know the basis about Schur-Complement. Anyway, while looking at this inequality to apply it in order to solve for $\lambda$ such that the the matrix is definite positive, I got a little bit confused ...
2
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1answer
60 views

Easiest way to show positive semi-definite equivalence

For an $x \in \mathbb{R}^n$, and $n$-by-$n$ identity matrix $I_n$, we are given that $$ \begin{pmatrix} I_n & x \\ x^T & 1 \end{pmatrix} \succeq 0.$$ What is the easiest way to show that $$ \...
0
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1answer
49 views

how to prove the following linear matrix inequality

$K=\{1,2,\dots,m\}$ $A_1,\dots, A_m$ be any square matrices. $n\times n$ say. If there exist positive definite symmetric matrices $P_1,\dots, P_m$, $(n\times n)$ say. such that, $A_i^TP_jA_i-P_i<...
2
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0answers
41 views

pole placement in LMI regions

In this paper, the authors elegantly present a LMI region as a subset $\mathcal{D}$ of the complex plane $$ \mathcal{D} = \{z \in \mathbb{C} | L + z\cdot M + \bar{z}\cdot M^T < 0 \}$$ where $L = L^...
1
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1answer
31 views

Can an upperbound constraint on the squared Frobenius norm of a matrix be expressed as a linear matrix inequality?

Is it possible to express an inequality constraint on the squared Frobenius norm of a matrix $X$: $$\|X\|_F^2 = \mathop{tr}( X^T X ) \le t$$ as a linear matrix inequality? I want to say that it's: ...
0
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1answer
58 views

Find diagonal matrix $D$ such that $A D$ is Hurwitz

Let $A \in \mathbb{R}^{m \times m}$. Give necessary and/or sufficient conditions for the existence of a matrix $D \in \mathbb{R}^{m \times m}$ such that all eigenvalues of $AD$ have negative real part ...
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1answer
53 views

How do I express this inequality as an LMI?

I have the following matrix inequality that I need to express as an LMI $ (AQ+BY) Q^{-1} (AQ+BY)^T - Q + \sum_i (A_i Q+B_i Y) Q^{-1} (A_i Q + B_i Y)^T < 0$ $Q > 0$ The matrices $A$, $A_i$, $B$...
2
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1answer
60 views

Can I apply the Schur complement on both sides of an inequality

I have the following matrix inequality that I need to express as an LMI $ A^T Q^{-1} A - Q + \sum_i A_i^T Q^{-1} A_i < 0$ ($Q > 0$, $A = XQ+YB$, $A_i = X_i Q+Y_i B$) which I rewrote to $ A^T ...
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0answers
27 views

Simplification of Linear matrix inequality

I have a question about simplification of Linear matrix inequality. I want to simplify below equation in to inequality form. \begin{align} A^TF_{1}^T(t)B^TF_{2}^T(t)C^TP + P C F_{2}(t)BF_{1}(t)A \...
1
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0answers
88 views

Linear matrix inequality

I have the following LMI (linear matrix inequality): $F^TP+PF\leq 0$ where $P$ is a positive definite matrix. if I have the following condition $Q\leq P$, $Q$ is a positive definite matrix. is it ...
1
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1answer
107 views

Searching an analogues for Schur complement

When I'm trying to solve a matrix inequality set of the following: \begin{equation} \begin{array}{l} A-BC^{-1}B^{T}>0\\ C>0 \end{array} \end{equation} Where $A$ is a given $p\times p$ ...
0
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1answer
71 views

Can I get a closed form solution of this SDP?

$$\begin{array}{ll} \text{maximize} & t\\ \text{subject to} & \mathbf{A} -t \mathbf{B} \succeq 0\end{array}$$ where $\mathbf{A}\succeq0$ and $\mathbf{B}\succeq0$. I want to ask one question. ...
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35 views

How to find a matrix with eigenvalues of different signs using LMI tools?

Suppose I am given two $n \times n$ real matrices $A_1$ and $A_2$ and I am wondering if there exists a positive definite matrix $P=P^\top$ such that (this is related to the stability of linear systems)...
0
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1answer
79 views

Question related to example 3.4 of Convex Optimization book by Boyd & Vandenberghe

In this example we have following inequality $$t-\mathbf{x^TY^{-1}x}\geq 0$$ which can be equivalently written in matrix form as follows $$\left[ {\begin{array}{cc} \mathbf{x^T} & 1 \\ \end{...
2
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2answers
104 views

Every convex polyhedron is a spectrahedron

I'm trying to show that convex polyhedra are special cases of spectrahedra. This was left as an exercise to the reader in a convex optimization text that I'm reading. I'm not sure how standard the ...
1
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0answers
54 views

Transforming matrix inequality into LMI

Is there a variable change I could do in order to transform the matrix inequality below $$A^T P A - P + Q + K^T B^T P A + A^T P B K + K^T(B^T P B + R) K \leq 0$$ into an LMI? My variables are ...
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1answer
44 views

How is this a linear matrix inequality?

In example 3.4 of Stephen Boyd & Lieven Vandenberghe's Convex Optimization, it is mentioned that the last condition of $$\text{epi}=\{(x,Y,t) \mid Y\succ 0, x^TY^{-1}x\leq t\}$$ is a linear ...
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252 views

Conversion of convex QCQP to standard form SDP

I have the following QCQP: $$ \underset{\mathbf{x}}{minimize} \quad \mathbf{x}^{T} H \mathbf{x} + \mathbf{p}^{T} \mathbf{x}$$ $$ subject \; to \quad \mathbf{x}^{T}Q_{i}\mathbf{x} + \mathbf{q}_{i}^{T}...
3
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2answers
111 views

Showing a matrix inequality

Can someone help me with showing the following Let $X$, $B$, and $C$ all be positive definite matrices. Show that the following inequality is true: $$ (X + C)^{-1} (X B X' + C ) ( X + C)^{-1} \ge (B^...
3
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1answer
75 views

Show LMI $F(x)\succ0$ is feasible if and only if the LMI $F(x) \succeq I_{n \times n}$ is feasible

Let $F : V \to \Bbb S^{n\times n}$ be a linear map, where $V$ is a vector space and and $S^{n\times n}$ is the set of $n \times n$ symmetric matrices. Prove that the LMI $F(x) \succ 0$ is feasible if ...
3
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1answer
115 views

Comparing positive definite matrices and their inverses [duplicate]

Given two $n\times n$ symmetric real positive definite (p.d.) matrices $A$ and $B$, it is known that $$A \prec B \iff B^{-1} \prec A^{-1}$$ where $A\prec B$ means that $B-A$ is p.d.. Suppose $A\prec ...
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0answers
128 views

Can a linear matrix inequality constraint transform to second-order cone constraint(s)?

Maybe the answer for the general case is NO but consider the following problem as maximizing the minimum singular value of a matrix: $minimize\,\,\,\,{\sigma _{\min }}$ $subject\,\,to\,\,{A^T}A - {\...
0
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0answers
45 views

Transform LMI problem into different SDP form [duplicate]

I am trying to transform a LMI problem of the following form: $ min_x \quad c^Tx \\ s.t. \quad x_1A_1+\dots+x_nA_n \preceq R$ into another SDP formulation: $ min_x \quad c^Tx \\ s.t. \quad Ax = b, \...
2
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1answer
79 views

Riccati inequalities solution via LMI

$${A^T}P + PA + 2{C^T}C + \frac{1}{{{\gamma ^2}}}P{G_1}G_1^TP + P{G_2}G_2^TP - PB{B^T}P < 0$$ I want to know how to calculate the positive definite $P$ through LMI. Thank you very much.
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114 views

Equality constraints represented as LMIs

I have three complex variables $$x=\begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}'$$ two complex knowns $$d=\begin{bmatrix} d_1 & d_2 \end{bmatrix}'$$ and two equality constraints ...
0
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0answers
157 views

Transform semi-definite programming problem with linear matrix inequality(LMI) to SDP of standard form

For a SDP problem with LMI, we can write it as: $$minimize \quad c^Tx$$ $$ s.t. \quad x_1F_1+x_2F_2+...+x_nF_n+G \preceq 0$$ $$Ax=b$$ where $G,F_1,F_2,...,F_n \in S^k,A \in R^{p \times n}$. Now I ...
1
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1answer
186 views

Strict Inequality in Homogenous LMI

I'm studying Stephen Boyd's notes for EE 363, here. In particular, I'm working through lecture 15, slide 9 on strict linear matrix inequalities. An LMI is an expression of the form $G(x) = G_0 + ...
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0answers
316 views

Writing an accurate SDP solver in Matlab

As part of a research project I'm supposed to write an semidefinite programming solver in Matlab (similar to SDTP3, MOSEK, SEDUM, etc) except it needs to be able to solve to many significant digits ...
1
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1answer
246 views

SOSTools - Getting example to work?

EDIT: Changed example to a more suitable one for discussion (smaller): I am trying to solve example 4.2 from this paper: Analysis of Non-polynomial Systems using the Sum of Squares Decomposition. ...
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0answers
205 views

How to transform the following inequality into LMI (linear matrix inequality)?

Let $B$, $C$, $W$ and $V$ be given (known) matrices with $V$ and $W$ being semidefinite positive. We would like to determine the matrices $X$, $Z$ and $T$ by solving the following inequality \begin{...
0
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1answer
72 views

How can we solve this matrix inequality?

Let $A$, $C$, $W$ and $V$ be given (known) matrices with $V$ and $W$ being semidefinite positive. We would like to determine the matrices $X$, $Y$, $Z$ and $T$ by solving the following inequality \...
1
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1answer
85 views

How can we solve the following matrix inequality?

Let $A$, $C$, $W$ and $V$ be given (known) matrices with $V$ and $W$ being semidefinite positive. We would like to determine the matrices $X$ and $Y$ by solving the following inequality \begin{...
4
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0answers
355 views

Infinity norm of matrix as inequality constraint in optimization

Imagine the optimization problem below, where the matrix $\ell_\infty$ norm appears as an inequality constraint: \begin{equation} \min_{X} \| XB - C\|_F^2 \\ \text{s.t. } \|AXB - B\|_\infty \leqslant ...
2
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1answer
81 views

LMI result for quadratic stability of norm-bounded differential inclusions

I am trying to understand how equation (5.14) is derived in the textbook Linear Matrix Inequalities in System and Control Theory by Boyd et al. Specifically, the result deals with the quadratic ...
1
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2answers
671 views

Algebraic Riccati Inequality Solution via LMI

I'm facing the following problem obtaining the solution of the Discrete Algebraic Riccati Inequality. Notation and assumption: $\succeq, \succ,\preceq,\prec$ refers to matrix definiteness ...
1
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1answer
68 views

Expressing (epigraph) inequality involving trace of matrix product as a matrix inequality

I want to express an inequality of the form $$\mbox{tr} (A^{-1}B)\leq t$$ as a matrix inequality, where $A$ is positive definite and $B$ is positive semidefinite. In particular, the matrix ...
2
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1answer
520 views

How to solve this matrix inequality?

Let $C$ be a given (known) matrix and let $\theta$ be a given (known) positive real. We would like to determine the matrices $X$ and $Y$ and diagonal matrix $P$ solving the following inequality \...
4
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1answer
1k views

Writing a convex quadratic program (QP) as a semidefinite program (SDP)

Given a convex quadratic program (QP) $$\begin{array}{ll} \text{minimize} & \mathrm x^\top \mathrm Q \, \mathrm x + \mathrm r^{\top} \mathrm x + s\\ \text{subject to} & \mathrm A \mathrm x \...
4
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1answer
557 views

Minimization of Frobenius norm and Schur complement

There is a famous problem here; however, this is not I want to ask (about proximal operator). Suppose I have an easy optimization problem: $$\min_Q \|Q-Q_N\|_F$$ where $\|\cdot\|_F$ is the ...
3
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1answer
260 views

The equivalence of $Y=XX^T$ - by Schur complement and rank constraint

I am confused about the following lemma which is useful to convex optimization problem: ( From http://ieeexplore.ieee.org/document/599549/ ) I know the left one is by Schur complement (https://en....
5
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1answer
741 views

Spectral norm minimization via semidefinite programming

Given symmetric matrices $A_0, A_1, \dots, A_n \in \mathbb R^{m \times m}$, let $A(x) := A_0 + x_1 A_1 +\cdots + x_n A_n$. How to formulate the following unconstrained spectral minimization problem as ...
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2answers
147 views

Reformulate linear program as semidefinite program

We consider the linear program $$\min_{x \in R^n} \{c^Tx \mid a_1^Tx \le b_1, a_2^Tx \le b_2\}$$ where $c, a_1, a_2 \in \mathbb R^n$ and $b_1, b_2 \in \mathbb R$ are given. Now we need to ...
3
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0answers
58 views

Set Containment

I have a quadratic, positive definite function $V(x)$ for which I am attempting to find a level set that is contained within two level sets of another quadratic, positive definite function. The ...
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0answers
212 views

Write the intersection of two quadratic inequalities as the union of quadratic inequalities

For any $m$ by $m$ matrix, $M$, I define the following corresponding set, $$ \mathcal{X}_M = \left\{x \in \mathbb{R}^m \,|\, x^T M\, x > 0\right\}. $$ Now for given are matrices $A$ and $B$ both ...
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1answer
57 views

How to compute an Linear Matrix Inequality region for a conical sector

I have to compute a matrix $P$ that defines an Linear Matrix Inequality region in the following way $$L_P = \{ s\in \mathbb{C}|\begin{pmatrix}I \\sI \end{pmatrix}^* P\begin{pmatrix}I \\sI \end{...
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0answers
124 views

From SDP to EVP

How to transform the semidefinite program (SDP) $$ \min \, \, c^T x $$ $$ \text{subject to}\,\, F(x)>0 $$ where $F>0$ is a LMI ($F(x)=F_0 +\sum_{i=1}^n x_iF_i(x)$, $F_i$ symmetric matrix) and $...
12
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2answers
497 views

What is the volume of the $3$-dimensional elliptope?

My question Compute the following double integral analytically $$\int_{-1}^1 \int_{-1}^1 2 \sqrt{x^2 y^2 - x^2 - y^2 + 1} \,\, \mathrm{d} x \mathrm{d} y$$ Background The $3$-dimensional ...
3
votes
1answer
358 views

Maximize $\langle \mathrm A , \mathrm X \rangle$ subject to $\| \mathrm X \|_2 \leq 1$

Given $\mathrm A \in \mathbb R^{m \times n}$, $$\begin{array}{ll} \text{maximize} & \langle \mathrm A , \mathrm X \rangle\\ \text{subject to} & \| \mathrm X \|_2 \leq 1\end{array}$$ ...