All Questions
Tagged with convex-optimization discrete-geometry
8 questions
1
vote
0
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79
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Touring a sequence of convex polygons with minimal energy
There is a known problem of touring a sequence of $n$ polygons: given a starting point $s$, an ending point $t$ and a sequence of polygons $P_1,\dots,P_k$ with a total of $n$ vertices, find points $...
- 177
4
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255
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Economic equilibrium and tropical geometry
There is a famous saying in economics: When everyone pursues his or her own interests, there is an invisible hand that brings the market to equilibrium. However, this is not always the case. Here is ...
- 123
1
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920
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Maximizing a piecewise-linear convex function
Crossposted on Operations Research SE.
I am working on an optimization problem where some of the terms of the objective function to maximize are expressed as a piecewise linear function of variables:
...
- 111
4
votes
2
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243
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Clustering of vertices in an $n$-dimensional cube
Consider the vertices of an $n$-dimensional cube. The distance between two vertices is measured as the minimum number of edges between the two vertices. Now consider a subset of these vertices.
If we ...
- 41
2
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62
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on vectors for which the intersection of their convex hull and the nonegative orthant is the unit simplex
Consider the vectors $r^1 = (0,2,-1)$, $r^2 = (-1,0,2)$, and $r^3 = (2,-1,0)$. Two properties of these vectors that interest us here are:
1) The $i$'th coordinate of $r^i$ is 0, and
2) The ...
- 745
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180
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Question on solving an optimization problem using Variable splitting and ADMM
Tell me if I have found the right approach to the following optimization problem:
$$
min_{x} \frac{1}{2}\left \| Ax-b \right \|_2^2
\\
s.t. \ \ \Phi v=x \ , \ {x^T(1-x)}=0
$$
$A$ and $\Phi$ ...
- 109
5
votes
1
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146
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How does one go from convexity to submodularity?
If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...
- 1,893
7
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1
answer
216
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How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?
This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...
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