Questions tagged [global-optimization]
The global-optimization tag has no usage guidance.
13
questions
37
votes
1
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Sofa in a snaky 3D corridor
What is the largest volume object that can pass though a
$1 \times 1 \times L$ "snaky" corridor, where $L$ is large
enough to be irrelvant, say $L > 6$.
...
15
votes
3
answers
1k
views
Is this lower bound for a norm of some complex matrices true?
Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$.
I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(...
1
vote
1
answer
250
views
Can we invoke "almost supermartingale" Theorem for deterministic sequences?
Perhaps stupid question.
Question: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems?
Attempt ...
18
votes
1
answer
625
views
Known configurations maximizing the volume of the convex hull of n points on the unit sphere
For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the ...
7
votes
0
answers
2k
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Optimization over space of probability measures
Consider an optimization problem as follows:
$$
\min\mathbb E_w[f_0(w)] \mathrm{\,\,\,\,\,\ s.t.\,\,\,\,} E_w[f_i(w)]\leq 0 ,\,\,\, i=1,\dots, k
$$
where the maximum is taken over $\mathscr M$, ...
7
votes
1
answer
396
views
Does the plane clustered to minimize sum distances^2 to clusters centers ( inertia / "K-means") produce hexagonal clusters / hexagonal lattice?
"K-means" is the most simple and famous clustering algorithm, which has numerous applications.
For a given as an input number of clusters it segments set of points in R^n to that given number of ...
6
votes
1
answer
2k
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 ...
2
votes
0
answers
131
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Can we conclude $\sup_g\int f_1g\le\sup_g\int f_2g$ from $\int f_1\le\int f_2$ in this situation?
Disclaimer: Please bear with me, the question isn't as complicated as it looks like, but I wasn't able to find any simplification for which no counterexample comes to my find.
Let $(E,\mathcal E,\...
2
votes
1
answer
573
views
Maximal minimum for a sum of two (or more) cosines
Please prove (or disprove, and give the correct answer):
$$2 =\mathrm{argmax}_{r\geq 1}\min_{x\in \mathbb{R}}\left[\cos\left(x\right)+\cos\left(rx\right)\right]
$$
In other words, find $r \geq 1$, ...
1
vote
2
answers
116
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How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?
I am looking for an algorithm to solve the following optimization problem
$$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$
where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$.
...
1
vote
1
answer
221
views
Maximize a Lebesgue integral subject to an equality constraint
I want to maximize $$\Phi_g(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)\int\lambda({\rm d}y)\left(w_i(x)p(x)q_j(y)\wedge w_j(y)p(y)q_i(x)\right)\sigma_{ij}(x,y)|g(x)-g(y)|^2$$ over all choices ...
1
vote
1
answer
189
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Proof of extended version of non-random "almost supermartingale"
In this question, a non-random version of "almost supermartingale" theorem is proved.
Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
0
votes
0
answers
235
views
Limit of argmin of sum
Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it ...