1
vote
1answer
42 views

Find max and min subject to constraint ||x|| = 4

$Q(x,y)=7x^{2}+12xy+12y^{2}$ I only know how to do this is $\|(x,y)\|=1$ If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$. I don't know what to do if ...
0
votes
0answers
18 views

Optimization with intervals

I am trying to solve a specific problem, and I was able to summarize it in the following optimization problem. I have a portfolio comprised of two assets. Asset 1 has return $r_1$, standard deviation ...
2
votes
2answers
66 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 ...
1
vote
2answers
50 views

steepest descent with quadratic form converge in 1 iteration

Well I'm stuck on an exercise given: The steepest descent method is applied to the quadratic form $$Q(\mathbf{x}) = \tfrac{1}{2}\mathbf{x}^TA\mathbf{x} - \mathbf{b}^T\mathbf{x} + c$$ where $A$, ...
1
vote
0answers
52 views

How to solve an optimization problem with non-convex Frobenius norm constraint?

The form of my problem is: $$ \min_W \|Y-WX\|_F^2-\|V-WU\|_F^2 $$ $$ s.t. \|W\|_F=1 $$ All five variables are matrices. Since the norm constraint is a non-convex one, I have no idea how to solve this ...
1
vote
1answer
119 views

Multivariable local maximum proof

Suppose we have a twice differentiable function $f: \mathbb{R} ^n \to \mathbb{R}$, a point ${\bf x^0} = (x_1 ^0 , \ldots , x_n ^0)$ and we know that $\nabla f({\bf x}^0) = 0$ $({\bf x - x^0})H({\bf ...
0
votes
0answers
35 views

Solving intersection of 4 quadratic equation with constraints

I want to find the intersection of 4 quadratic equation with the constraint that $\left[\begin{matrix}q_0 \\ q_1 \\ q_2 \\ q_3 \end{matrix} \right]$,$\left[\begin{matrix}p_0 \\ p_1 \\ p_2 \\ p_3 ...
0
votes
1answer
62 views

Extremum of a multidimensional quadratic function

I have the following function: $$ g(h) = h'\Sigma\Sigma'h-h'm-r, $$ where $h$ is a vector in $\mathbb{R}^M$, $\Sigma$ is a $M\times K$ matrix such that $\Sigma\Sigma'$ is positive definite and has ...
2
votes
0answers
90 views

Is there a generalization of eigenvalues/eigenvectors to $L^p$ for $p\neq2$?

Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem: $$ \min_{v : \left\|v\right\|_p \ge c} ...
0
votes
1answer
33 views

Optimization of Unconstrained Quadratic form

So I'm learning about optimization of quadratic forms and this textbook goes through definiteness of matrices and principle minors etc. and then goes straight onto optimizing with constraints but ...
2
votes
1answer
152 views

A question regarding local minimizer of a function restricted on a circle

I have a quadratic function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f(\mathbf{x}) = (\mathbf{x}-\mathbf{p})^\top \mathbf{Q} (\mathbf{x} - \mathbf{p})$ where $\mathbf{Q}$ is positive definite and ...
2
votes
3answers
513 views

positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than 1 and off-diagonal entries positive but strictly less than 1 necessarily positive-semidefinite?
1
vote
1answer
213 views

Finding the minimum value of a quadratic within a range

Given any quadratic equation of the form $y=ax^2+bx+c$, I want to find the minimum value for a specific range of $x$. My programmer brain can do it in a branchy, algorithmic way as follows, but is ...