2
votes
1answer
47 views

Can this multidimensional non-linear equation with constraints be minimized analytically?

I wish to find the vector of real numbers, $\mathbf{w}$, that minimizes the function: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
2
votes
1answer
43 views

Least squares problem with orthonormality constraints

Given $y_1,\ldots,y_n\in \mathbb{R}$,$w\in \mathbb{R}^d$, and $x_1,\ldots x_n\in \mathbb{R}^D$, how do we solve the following optimization problem \begin{align} \min_A \sum_{i=1}^n (y_i-w^TA^Tx_i)^2\\ ...
0
votes
0answers
24 views

Non-linear, non-convex optimization problem

I would like to solve the following optimization problem: $$ \max\{\min_{k=1,2,...,m}\{R_d^k\}+\sum_{k=1}^m R_e^k\} \\\text{s.t.} \\ \mathrm{trace}(W_d^kW_d^{kT})=1 \\ \mathrm{trace}(W_e^kW_e^{kT})=1 ...
1
vote
0answers
33 views

Decomposition of a symmetric semi-definite matrix into sums of sparse symmetric semi-definite matrix

I'll first provide the background. Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables $x^{(1)},\ldots,x^{(n)}$. We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
0
votes
1answer
260 views

Positive semidefinite Matrix examples query

This might be really dumb question but I've just started dealing with such matrices. I would like to know why $$\begin{bmatrix} 0 & 1 \\ 1 & x \end{bmatrix}$$ cannot be a positive semidefinite ...
0
votes
1answer
112 views

Minimization to Maximization doubt in SVM

I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done... $ Max \frac {1}{||x||} $ ...
2
votes
2answers
82 views

A Quadratic Problem (which looks very simple)

This arises as a part of my work. \begin{align} \min_{x^{H}x=1}~&x^{H}A_1x \\ subject~to~&x^{H}A_2x=0 \end{align} $A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
1
vote
0answers
378 views

Significance of Rank of Jacobian

I have been struggling with this question. What is the significance of finding the rank of a Jacobian matrix of a function? I understand that the Rank of a matrix signifies the number of linearly ...
2
votes
1answer
141 views

Positive values for a set of quadratic forms of Hermitian Matrices. (To find a set of vectors in which a hermitian matrix is positive definite)

Assume all matrices I discuss about are $N \times N$ and the vectors conform with dimensions. Consider the following set of Quadratic inequalities where all the matrices $A_i$ are hermitian. ...
1
vote
5answers
231 views

Independence of Rotation Matrix Definitions

I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix $$ A = \left( \begin{matrix} a_{11}& a_{12}& a_{13}\\ a_{21}& ...
2
votes
2answers
882 views

Modified Cholesky factorization and retrieving the usual LT matrix

I have been looking at the modified Cholesky decomposition suggested by the following paper: Schnabel and Eskow, A Revised Modified Cholesky Factorization Algorithm, SIAM J. Optim. 9, pp. 1135-1148 ...