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10
votes
0answers
188 views

“Small” maps from sphere to sphere

Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now ...
5
votes
0answers
360 views

Maximizing the matrix norm

Hi all, I wish to find a 3x3 rotation (orthogonal) matrix $\mathbf{R}$ such that it maximizes the following matrix norm: $||\mathbf{A} \mathbf{D}_r \mathbf{D}_b ||_2$ where $\mathbf{A}$ is a known ...
3
votes
0answers
137 views

Generalization of the equilateral triangle ?

I consider points in the two-dimensional plane. An equilateral triangle is a set of three points in the plane which are equidistant. Suppose now I have $n$ points $x_1,...,x_n$. What is the ...
3
votes
0answers
309 views

Minimum weight bipartite graph clique covering

I was wondering if anyone here could give me any pointers as to how to solve the following problem. Let $B=(L,R,E)$ be a bipartite graph, and $\forall u\in L\cup R$, let $c_u$ be a cost associated to ...
2
votes
0answers
160 views

Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with $m-$dimensional coordinates in ...
2
votes
0answers
154 views

modification of singlestart in global optimization

When minimizing a nonconvex function $f : \Omega \rightarrow \mathbb{R}$ that may have multiple minima, there are some very simple strategies to improve the odds of finding the global minimum point. ...
1
vote
0answers
58 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
1
vote
0answers
76 views

Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...
1
vote
0answers
114 views

Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function. Also, the member of $X$ are uniformly bounded $$ |x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}. $$ Let $F: X ...
1
vote
0answers
39 views

What is the sufficient condition for a “strict local optimal point” to be “isolated local optimal point”(or strong local optimal point)?

I encountered a case that seems obvious that the local optimal point are isolated, yet I can only prove the local optimal points are strict rather than isolated. I know under certain peculiar ...
1
vote
0answers
94 views

Trying to get an idea of the maths I could use for this optimization problem

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...
1
vote
0answers
121 views

An $L^{\infty} Version of Principal Component Analysis?

I have a $k$ by $n$ matrix $A$, with $k \ll n$. In case it helps, the $k$ rows are orthonormal. I'm interested in finding a $k$ by $k$ orthogonal matrix $M$ so as to maximize the $L^{\infty}$ norms ...
1
vote
0answers
168 views

On the convergence of a special fixed point iteration

The problem is actually a quadratically constrained quadratic program. And the formulation is: $max: \frac{1}{2}x^TQx + d^Tx$ $s.t. x\in R^{n,+} ,\sum_{i\in I_p}x_i^2=1, p=1..k$ where $d\in ...
0
votes
0answers
53 views

McCormick relaxation of nonsmooth functions

I was wondering if we can use a McCormick based relaxation for non-smooth functions that have besides intrinsic functions, binary addition and binary multiplications, are composed of other non-smooth ...
0
votes
0answers
75 views

The role of subgradient in programming with nonsmooth functions

It is obvious that there is similarity between subgradient and gradient. The subgradient of smooth functions is reduced to gradient. I have two questions. The first is does there exist subgradient ...
0
votes
0answers
54 views

equivalence between primitive and dual

Hi everyone! I have a problem about the duality gap of the primitive problem and the dual problem. This problem comes from a probabilistic model named Lagrangian UVM. ...
0
votes
0answers
360 views

optimization of a separable function

Hello everyone, this is a optimization problem whose objective function is separable: $$F(x)=\sum_{i=1}^n\frac{\theta_i^2}{4}\sum_{j=1}^m\left(1+\rho ...
0
votes
0answers
247 views

Optimization of a matrix with an objective function (for ML)

Hi. I need to do max. likelihood for an objective likelihood function L (minimize it), and the target is a matrix. ie: $$min_KL(K)$$ For example: K is, let's say, of size 3x3 and with initial ...
0
votes
0answers
432 views

A product sum inequality question

For any $x_{1},x_{2},\cdots x_{6}$ with $\sum_{i=1}^{6}x_{i}^{2}=1$ and $y_{1},y_{2},\cdots y_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}y_{i}^{2}=1$, do there always exist $z_{1},z_{2},\cdots z_{6}$ in ...
0
votes
0answers
91 views

minimizing the integral of a function over square sets.

Hi! I'm interested in some problems, but to be honest i'm not sure of the field they belong to. Let $h(x,y)$ be a bivariate function on $X^2$, where $X$ is some nice topological space (for instance ...