**21**

votes

**5**answers

1k views

### When does symmetry in an optimization problem imply that all variables are equal at optimality?

There are many optimization problems in which the variables are symmetric in the objective and the constraints; i.e., you can swap any two variables, and the problem remains the same. Let's call such ...

**7**

votes

**4**answers

1k views

### Is there a name for the matrix equation A X B + B X A + C X C = D?

I happen to be working on a problem that reduces to solving the following equation:
$$\mathbf{A X B} + \mathbf{B X A} + \mathbf{C X C} = \mathbf{D}$$
where A through D are known matrices ( A, B, D ...

**5**

votes

**5**answers

1k views

### Application of polynomials with non-negative coefficients

Question 1: Are there any deeper applications (in any field of mathematics) of polynomials (with possibly more than one variable) over the real numbers whose coefficients are non-negative? So far I ...

**11**

votes

**4**answers

444 views

### Minimal-length embeddings of braids into R^3 with fixed endpoints

(Apologies in advance for any imprecision in the following; I am a computer scientist and regret never having taken an actual course on topology.)
One way to define the pure braid group $P_n$ is as ...

**3**

votes

**4**answers

1k views

### Minimizing the modulus of a polynomial around a circle

I'm probably missing something elementary here, but I guess the only way to be sure is to ask here.
Now, I have encountered a situation where given an nth-degree polynomial $p_n(z)$ with complex ...

**5**

votes

**1**answer

888 views

### Non-negative quadratic maximization

For a given symmetric and positive semidefinite $n \times n$ matrix $A$, we want to solve the problem $$\max_{||x|| = 1, \ x\geq 0} x^T A x.$$
Here, $x\geq 0$ indicates that $x$ must be component-wise ...

**6**

votes

**2**answers

778 views

### Conic hulls and cones

Suppose I have a number of vectors in $\mathbb{R}^n.$ The first question is: what is the most efficient algorithm to compute their "conic hull" (the minimal convex cone which contains them)? The next ...

**4**

votes

**0**answers

278 views

### The Gömböc and monostatic objects

This made the popular news a few years ago. Summary: it's the first homogeneous, convex solid to be found with only one stable and one unstable mechanical equilibrium when resting on a flat surface. I ...

**4**

votes

**2**answers

354 views

### Simplified knapsack problem

There is a problem that I can not solve.
Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...

**3**

votes

**1**answer

437 views

### Symplectic block-diagonalization of a real symmetric Hamiltonian matrix

Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...

**3**

votes

**2**answers

481 views

### Cubic graphs which are “difficult to navigate”

Suppose I am inside a finite, weighted cubic graph without loops, with no information regarding its layout, including the number of vertices or distances to the adjacent vertices. I want to reach a ...

**2**

votes

**1**answer

96 views

### Optimizing finite-length approximations to space-filling loops

Take a loop in the unit disk D^2, with length l where length is defined as the supremum of the lengths of piecewise linear approximations. What is the smallest r such that every radius-r subdisk of ...