1
vote
1answer
21 views

Properties determining boundedness of function

The function I am looking at is $$f(x) = \frac{1}{2}x^TAx + b^Tx + c$$ where $A$ is a symmetric matrix in $\mathbb{R}^{n\times n}$ and $b,c$ belong to $\mathbb{R}^n$ I want to determine what ...
1
vote
1answer
29 views

Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$

The following question is similar to this one, but I think that it is not straightforward to move from one to the other, so please take a look. Otherwise, please let me know and I will delete it. ...
0
votes
0answers
23 views

how to prove this sparse coding equation

How can I prove the following? $\sum_i \frac{1}{2} \|\mathbf{x}_i - D\mathbf{\alpha_i}\|^2 = \frac{1}{2}Tr(D^TDA_t) - Tr(D^TB_t)$ where, $A_t = \sum_{i=1}^T \mathbf{\alpha}_i\mathbf{\alpha}_i^T\\ ...
0
votes
1answer
27 views

Optimising using Hessian matrix

I am bit perplexed in optimisation problem if the principal minor is zero. If the principal minor is zero does it mean that the Hessian matrix is always indefinite and the point of extremum will refer ...
0
votes
0answers
18 views

Which matrix norm gives the minimal variation of eigenvalues?

This is a follow-up of this question. The original question is intentionally as general as possible, because I was interested in the most general possible answer. I am now trying to understand its ...
0
votes
1answer
21 views

Comparing two circularly shifted matrices

I am looking for a way to compare two matrices A and B where B is the result of circularly shifting rows of A i.e. A = [1 2 3;4 5 6], B = [4 5 6;1 2 3] Is there an operator or metric that would ...
0
votes
0answers
20 views

portfolio optimisation

I'm currently implementing a CAPM model in Excel based on the following criteria/features: A portfolio of n risky assets when n=6 (in this case) A riskless borrowing rate of 8% and riskless lending ...
3
votes
2answers
64 views

finding the closest matrix of a given form

let's say I have a vector $(a_1\dots a_n)$, where each component is between $-1$ and $1$. Now from this vector I define a $n\times n$ matrix $M$ such that $$M_{ij} = \begin{cases} 1&\,& i = ...
1
vote
1answer
83 views

Proof of a matrix is positive semi-definite

For $\ i = 0, 1, \cdots m$, $f_{i}(x): R^n \rightarrow R$ is defined to be $$ f_i(x) = x^TQ_ix + 2p_i^Tx + r_i $$ , where $Q_0 \cdots Q_m$ are real symmetric matrices, $p_0 \cdots p_m \in R^n$, and ...
0
votes
0answers
27 views

resources about sparse global constrainted optimization

Please recommend a good resources (books/articles/software) about sparse global constrained optimization?
3
votes
1answer
44 views

Trace minimization subject to constraints

I have seen in an article that $ \min_{\mathbf{K}} \hspace{0.2cm} tr[\mathbf{K} \Sigma \mathbf{K}^T]$ s.t. $ \mathbf{KH} = \mathbf{I} $ where $\mathbf{H}$ is of full column rank yields, ...
1
vote
1answer
78 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
1
vote
2answers
100 views

Unsolvable(?) Assignment Problem

I've recently been trying to implement the Hungarian Method in C++, and I've been using 5x5 matrices to test my program. Last night I came across a matrix which neither I nor my program can solve. Is ...
1
vote
1answer
36 views

Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with. Can I claim ...
0
votes
0answers
29 views

Homography between known and unknown rectangle corners

I would like to know if there is a solution for the problem of homography estimation in the special case in which one of the views is unknown but has some constraints, particularly if we know the ...
2
votes
0answers
27 views

Random Rotation of Points using Householder matrices

I have $N$ points in $D$ dimensions, were $D$ is big, for sure more than $100$. $N$ is also big. The goal is to produce an algorithm in my code, that will take as input this dataset and will give ...
1
vote
0answers
23 views

Optimal VCV matrix solution of multivariate loglikelhood

I asked a related question yesterday and got a brilliant answer from Ross B. However I still have difficulties. I have the following analog of multivariate loglikehood function (minus 2*log-likehood ...
0
votes
0answers
51 views

Finding minimum of a distance function using matlab

I have a function for that I want to find the minimum. The function calculates the distance between two sets where a set is defined as matix of row vectors $ D = [ d_1, d_2, ..., d_n]$, $d_n$ is a $m ...
2
votes
2answers
82 views

If the Jacobian matrix is positive definite, does that imply that the optimization problem has a unique solution?

My PhD adviser told me that if the Jacobian matrix of the optimality conditions is positive definite, then it implies that the optimization problem has a unique solution. I was wondering what is the ...
1
vote
2answers
131 views

Trace minimization of a matrix

Suppose $S = \pmatrix{1&1\\ 1&0\\ 0&1}$, $W$ is a $3\times3$ covariance matrix, which could be regarded as fixed. I need to find a $2\times 3$ matrix $Q$ that minimizes $$ ...
0
votes
1answer
25 views

Can one determine optimal parameters of a matrix to design the matrix kernel? (with specific example)

Suppose we are given a matrix $B\in \mathbb{R}^{n\times n}$. I would like to find $n$ real values $\{a_i\}_{i=1}^n$ that form a diagonal matrix $A=\text{diag}(\{a_i\}_{i=1}^n)$ to design the kernel of ...
2
votes
1answer
54 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: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
3
votes
1answer
71 views

Rank one plus diagonal matrix approximation

Given $A \in R^{n \times n}$, $A$ symmetric. I'm trying to solve the following minimization problem: $\underset{u \in R^n, d \in R^n} \min \, \frac{1}{2} \|X - A\|_F^2$ subject to $X = u u^T + ...
1
vote
1answer
14 views

the differences and relationship between linear independent and affinely independent

When learning optimization, I heard the two related concepts on linear algebra: linearly independent and affinely independent. ...
0
votes
0answers
24 views

Convex matrix inequality

Consider a matrix inequality as $M(a,b,c)<0$ where $a>0$, $\bar b>b>0$ and $c\in[-1, 1]$. The problem is feasibility of this inequality for all the possible values of the parameters. Can I ...
1
vote
2answers
51 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
2
votes
1answer
91 views

Gradient of matrix exponential function

Grateful if somebody could help me with the following. I am trying to find the gradient of the next expression: $$f(a_1, a_2, a_3, a_4)=\Vert R*y-x \Vert $$ where $y$ and $x$ are known 4x1 column ...
2
votes
2answers
44 views

Minimizing Frobenius norm for two variables

I need to minimize squared Frobenius norm: $\|\mathbf{A} - \mathbf{x}\mathbf{y}^T\|_F^2$. Namely I need to prove that for this norm to reach minimum $\mathbf{x}$ should be eigenvector of ...
0
votes
2answers
61 views

How to find center and radius of hand-drawn circle? [duplicate]

You are given a set of points {(X1,Y1), (X2,Y2),...} which represent a hand-drawn circle, so it's not perfect. You are asked to find the center and radius of this circle. My intuition tells me this ...
1
vote
0answers
26 views

Matrix multiplication in game theory doesn't add up? Min y^T*Ax

I'm studying game theory and something seems weird to me. My book says y is the probability of the row player and x is the probability of column player, both x and y are vectors. A = [a$_i$$_j$] is ...
0
votes
0answers
22 views

Constructing Matrix with Normal Distribution

I have a vector given whereby each element of the vector is assumed to be the average of one of a matrix' rows. Now I want to construct the matrix belonging to this vector, whereby the elements of the ...
2
votes
3answers
46 views

How to combine Unitary Matrices in a clever way?

I am trying to implement genetic-type algorithms on unitary matrices. Hopefully I should be able to use this question for the mutation part. But I am having an issue with the cross-over step. So here ...
0
votes
0answers
15 views

Which cut-off for collapsing this tree?

I have a Newick tree that is built by comparing similarity (euclidean distance) of Position Weight Matrices (PWMs or PSSMs) of DNA regulatory motifs that are ~5-9 bp long sequences. An interactive ...
2
votes
1answer
55 views

Givens rotation and retraction mapping

Here's part of the texts in the book Optimization Algorithms on Matrix Manifolds, when talking about the retraction on a matrix sub-manifold. A retraction is a mapping that maps a tangent vector in a ...
1
vote
2answers
49 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
0
votes
1answer
63 views

Symmetric Positive Definite Matrix Proof

Suppose that $H^+ = H - (\mathbf y^TH \mathbf y)^{-1} H\mathbf y \mathbf y^T H + (\mathbf y ^T \mathbf s )^{-1}\mathbf s \mathbf s^T $ where H is symmetric and positive definite. Supposing that ...
0
votes
0answers
39 views

l1 minimization with orthogonality constraint

I want to find a rotation (or reflection) for my data which maximizes the space between my points and the basis' margins. I have formulated the problem as follows: Given $X \in \mathbb{R}^{n \times ...
2
votes
2answers
72 views

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
2
votes
0answers
48 views

Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
1
vote
0answers
69 views

Maximizing the product of first Eigenvalues of rank-1 hermitian matrices

Suppose we have $L$ complex vectors $\mathbf{a}_{l}$ with dimension $N\times 1$ I want to solve this optimization problem $\mathbf{x}_{\mathrm{opt}}=\arg ...
0
votes
1answer
185 views

How to calc $\min ||J\Delta\tau + D||_*$

How to calculate $$ \min_{\tau} ||J_1 \tau_1 + \cdots + J_p \tau_p + D ||_* $$ where $\tau_1, \cdots, \tau_p \in \mathbb{R}$ $J_1, \cdots, J_p, D \in \mathbb{R}^{m \times n}$ $||\cdot||_*$ is sum ...
0
votes
2answers
85 views

Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
0
votes
2answers
87 views

Minimize Frobenius norm with constraints

As a follow-up on my previous question, I would like to solve the following optimization problem: $\min \Vert MA-B \Vert_F^2-x^HMy\;\;s.t.\;\;M^HM=I$ where $A$ and $B$ are $N\times L$ complex ...
0
votes
1answer
43 views

A simple optimization problem

$$f = x^Tx$$ $$g = Ax-b $$ The constraint is $Ax-b = 0$ I calculated $J' = f'+\lambda g'$ which is $2x^T+\lambda A^T = 0 $ and $Ax-b=0$ . I dont know what to do next please help me out .
4
votes
1answer
106 views

Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
0
votes
1answer
82 views

Largest eigenvalue of symmetric matrix

I am trying to understand why the $\lambda_{\max}$ function is convex given an $n\,x\,n$ symmetric matrix, let's call it $A$. I know from elementary property of eigenvalues that all the eigenvalues of ...
2
votes
1answer
67 views

A Matrix Optimization Problem

Given an $n\times d$ matrix $Y$, I am looking for an algorithm to find an $n$-vector $\mathbf{v}$ ($0\le \mathbf{v}_i\le 1$ for all $i$) that minimizes $\sum_{i:X_i<0}X_i$, where $X:= \mathbf{v} ...
0
votes
0answers
18 views

Matrix Optimal Strategy Problem

(B) What is the expected value of the game for R if the bank R always chooses TV and bank C uses its optimum strategy? E= _ (type fully reduced fraction or mixed number) (C) What is the expected ...
1
vote
1answer
118 views

How to show this algorithm on positive semidefinite matrices converges to a global maximum determinant

I'm dealing with an algorithm which is supposed to converge to the maximum determinant of certain positive semidefinite matrices. The problem is that we have such a matrix, and we vary certain ...
1
vote
1answer
74 views

Minimum Eigenvalue of the Rank One update to a Positive Semi-Definite matrix

Let $\mathbf{A}$ be a $N\times N$ positive semi-definite hermitian matrix. Let $\mathbf{b}$ be a $N\times 1$ complex vector. For any given constant $t$, I interested in the minimum eigenvalue of the ...