Questions tagged [global-optimization]

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37 votes
1 answer
2k views

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 $L > 6$.           ...
Joseph O'Rourke's user avatar
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(...
Mahdi's user avatar
  • 1,991
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 ...
user550103's user avatar
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 ...
Gro-Tsen's user avatar
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7 votes
0 answers
2k views

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 $\mathscr M$, ...
Arash's user avatar
  • 722
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 ...
Alexander Chervov's user avatar
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 ...
user11443's user avatar
2 votes
0 answers
131 views

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,\...
0xbadf00d's user avatar
  • 161
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 $r \geq 1$, ...
Daniel Soudry's user avatar
1 vote
2 answers
116 views

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 $\mathbf{x}_i\in\mathbb{R}^d$. ...
user3750444's user avatar
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 ...
0xbadf00d's user avatar
  • 161
1 vote
1 answer
189 views

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 ...
user550103's user avatar
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 ...
ABIM's user avatar
  • 5,019