Operations research, linear programming, control theory, systems theory, optimal control, game theory

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21
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5answers
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
4answers
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
5answers
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
4answers
445 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
4answers
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
1answer
956 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
2answers
817 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
0answers
283 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
2answers
375 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
1answer
458 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
2answers
486 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
1answer
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 ...