# Tagged Questions

93 views

### Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...
157 views

### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation: $X=c \cdot AXA' - diag(c \cdot AXA')+ I$, where (1) $A \in R^{n \times n}$ is a given matrix whose element ...
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563 views

### solving Lyapunov-like equation

The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
318 views

### My overdetermined linear system gives both bad and good estimates. Why ?

Hello to everyone. What the question means is that different ways of expressing the same relation between the data and unknown variables produce really weird fit results: The problem: I have the ...
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Hi all -- what would be good methods (and/or software packages) to try for solving a problem minimizing a quadratic function $f(x) = \sum_{i=1}^N{(x_i - y_i)^2}$, where some constraints are non-linear ...
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I have the following quadratic optimization problem: $\min_{\vec{x}} |\vec{x}|^2$ subject to $\vec{x}^T G_j \vec{x} \geq 1$, $j = 1 \ldots m$, where the $G_j$ are positive semidefinite. $|\vec{x}|$ is ...
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### Projection onto a quadratic cone?

Consider a constraint of the form $$f(x) := x^T A x = 0$$ where $A \in \mathbb{R}^n$ is symmetric but may be singular and indefinite. The constraint set $C$ is a (nonconvex) cone, since for any ...
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### Relations between a set of inner products of vectors

Suppose we have n normalized vectors on an arbitrarily large Hilbert space $|A_1\rangle,\dots,|A_n\rangle$, $\langle A_i|A_i\rangle=1$ for every i. And there're $\frac{n(n-1)}{2}$ inner products ...
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### Decomposing max-convolution of sum of functions ?

Hello. $R, F_1, F_2, F_3$ are random (not-convex, not-concave) 2D matrices of size 100x100. $R$ is a linear combination of $F_1, F_2, F_3$. Explicitly, $R = w_1 F_1 + w_2 F_2 + w_3 F_3$ where ...
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### How do I optimize over (or take derivative wrt) a square diagonal matrix?

Hello. I'd like to solve the following optimization problem. $P_i$ is a 6x6 matrix $X$, $Y$ is a 6xk matrix $w_i$ is a kx1 vector $diag(w_i)$ is a square diagonal matrix with diagonal entries equal ...
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### Seeking proof for linear algebra constraint problem.

Given a symmetric real matrix with a zero diagonal $M$, I am trying to find a diagonal matrix $D$, such that the matrix $M + D$ is positive definite, and $(M+D)^{-1}$ has a diagonal consisting of all ...
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### Numerical linear algebra: how to compute $b^TA^{-1}b$ efficiently

What is the most efficient way to compute $b^TA^{-1}b$ for a given $A$ and $b$? Do we have to calculate $A^{-1}b$, or is this not necessary? edit: I forgot to mention that A is symmetric and ...
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### Problems finding feasible points with respect to linear matrix inequalty constraints

Hi! I'm a trying to learn the basics of semidefinite programming and how to solve problems with linear matrix inequalities. Inspired by the book "Convex Optimization" by Boyd and Vandenberghe (can be ...
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### An optimization problem in numerical linear algebra

Provided two diagonal real matrix which has positive entries, $V$ and $U$. Find a real matrix $A$, satisfying $A^TA=a^2I$ for some scalar $a$, to minimise $\left|A^TVA-U\right|\quad\quad(*)$ ...
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### Second order Taylor expansion to solve system of equations

Suppose you need to solve $f(\mathbf{x})=\mathbf{0}$ where $f:\mathbb{R}^n \to \mathbb{R}^m$, $m,n>1$. Newton's method relies on first order Taylor expansion of f. Where can I find details of ...
686 views

### Optimizing over matrices with spectral radius <1?

Suppose $F(x)$ is a convex objective function on $n\times n$ matrices, and I need to numerically optimize $F$ with the condition that $x$ has spectral radius less than $1$. This might be too hard, so ...
686 views

### Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs

Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
700 views

### Determine noise distribution [closed]

I'm trying to solve the following least squares problem: $\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$ where $Ax = b$ and $\tilde{b} = b + w$ Question: How do I determine which probability ...
627 views

### Condition number for Ellipsoid method matrix

Hello, When using the ellipsoid method (for solving a linear program for example), the volume of the ellipsoid at each iteration is proven to decrease, and do so by at least a factor of $e^{1/2n}$. ...
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### An optimization problem for points on the sphere (master's dissertation)

First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
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### Maximize the multiplicity of an eigenvalue

Hi, We have a real, non-singular and symmetric matrix M of size n by n, with diagonal elements 0's. Its eigenvalues and eigenvectors are computed. Now we wish to change its diagonal elements ...
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### Recommendations for a large scale bounded variable least squares (BVLS) solver for sparse matrices

I'm trying to solve the BVLS problem for huge (2e6x2e6) matrices which are very sparse (4 elements per row). Does anybody have a recommendation for a free solver (preferably a library of routines)? ...
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### minimize the sum of absolute eigenvalues

Hi, We have a real symmetric matrix M with diagonal elements 0's, the eigenvalues and eigenvectors of M are computed. Now we wish to change its diagonal elements arbitrarily to minimize the sum of ...
311 views

### Convergence to a (unique?) fixed point?

Consider a given $N\times P$ matrix $X$ (full rank with columns ${\bf x}_p$, $p=1,\ldots,P$), a given vector ${\bf y}\in R^N$ and a thresholding function $\eta_\lambda(|x|)=(|x|-\lambda)_+$ with ...
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### Nonlinear conjugate gradient update strategy by Dai and Yuan

In Nocedal and Wright book "Numerical Optimization", they describe on page 123 (formula 5.49) an update strategy for the beta parameter in the nonlinear conjugate gradient optimization, which was ...
1k views

### Finding the local/global minima of Shubert function

Consider the 2D Shubert function. As given in that page, the function has 18 global minima and several local minima. How can I find the (x,y) of all these minima? Any help appreciated. If it was a ...
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### Which method to apply to this problem?

I'm a programmer and I came a across an interesting problem. I'm sure there is a mathematical method or an algorithm to solve it, but I don't know where to start with the search nor which literature ...
1k views

### Minimizing a function containing an integral

I am trying to optimize a function of the following form: $L = \int_{t=0}^{T}(AR-x)dt$, where A is a system parameter i.e. I am trying to find the optimum x(t) that minimizes L over all admissible ...
Consider the optimization problem $$\min_x ||Ax||_1 + \lambda||x-b||^2,$$ where $A \in \mathbb{R}^{n \times n}$, $x,b \in \mathbb{R}^n$ and $\lambda$ is strictly greater than 0. (This problem is ...
I happen to be working on a problem that reduces to solving the following equation: $$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$ where A through D are known matrices ( A, B, D ...