0
votes
0answers
15 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
52 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
77 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
43 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
77 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
89 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
30 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
24 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
25 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
47 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
58 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
127 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
23 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
49 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
64 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
49 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
83 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
39 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
48 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
23 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
44 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
14 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
52 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
46 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
62 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
38 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
71 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
47 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
68 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
80 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
76 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
41 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
98 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
77 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
66 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
17 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
112 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
71 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 ...
0
votes
1answer
38 views

Sample Variance in Principle Components Analysis

I was reading this Why is the eigenvector of a covariance matrix equal to a principal component?. And in the top answer, the poster mentions that if the covariance matrix of the original data points ...
0
votes
1answer
68 views

Positive, Negative definite and indefinite matrix

A symmetric matrix is positive definite iff all eigenvalues are positive. I have been given a 3X3 symmetric matrix. I have calculated the eigenvalues two of which are negative. Does this mean this ...
0
votes
0answers
82 views

Minimizing the Kullber-Leibler divergence between two multivariate normal distributions

Take two zero-mean multivariate normal distributions: $p=\mathcal{N}(\mathbf{0},\boldsymbol\Sigma)$ and $q=\mathcal{N}\left(\mathbf{0},\left(\mathbf{A}^{T} \boldsymbol\Omega ...
1
vote
1answer
92 views

Finding minimum of the trace of the matrix equals finding maximum of the trace of the inverse matrix?

Let $K$ be a positive definite, symmetric matrix. Let $C$ be a nondegenerate matrix of the same order. Elements of $K$ and $C$ depend on some parameter $a.$ Is it true to say that $$ ...
1
vote
0answers
129 views

Area optimization: Packing rectangles inside rectangle

Background: I am scrambling to figure out the optimization algorithm to build sprite image, which is essentially a big container rectangular image, with multiple rectangular images. I have found an ...
3
votes
2answers
63 views

Matrix which when multiplied, gives a maximal minimum of elements of result.

I'm working on an optimization problem and am stuck at this particular step. Let $\bf{A}$ be a matrix with 4 columns and a finite number of rows, consisting of elements which are either 0 or 1. Let ...