# Tag Info

Accepted

### Illustrating that universal optimality is stronger than sphere packing

In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered ...
• 16k
Accepted

Accepted

### Maximal minimum for a sum of two (or more) cosines

So we start with a rational $r=\frac{p}{q}$. 1) Say $2\not|p,2\not|q$. Then taking $x=q\pi$, we get: $$f_r(x)=\cos(x)+\cos(rx)=\cos(q\pi)+\cos(p\pi)=-2$$ so $r$ is not the argmax. 2) Say $2$ divides ...
• 4,291
Accepted

### How to fit the parameters of differential equations with known data?

This kind of problem is known in the literature as a nonlinear state-space system identification. Several algorithms have been proposed in the literature to solve these problems. I think a good ...
• 1,494
Accepted

### Differences between the convex discrete maximization and minimization problems?

For continuous problems, minimizing a convex function on a convex domain is considered an easy problem, because there is only ever one local minimum, and a local minimum is the global minimum. ...
• 51.7k

### Can you give me good examples of non-convex functions that are problematic for optimization?

A less-known example is $f(x):=x^2+\exp(-1/(100(x-1))^2)-1$ on the closed interval $[-2,2]$. It takes $-.0067419337989203$ at $x = .996387676055289$. See that discussion in MaplePrimes for more ...
• 3,142
Accepted

### Maximizing the $\alpha$-moment of a distributution

Let us consider the closely related problem: maximize $EX^\alpha$ over all nonnegative random variables (r.v.'s) $X$ with $EX=\mu$. To avoid trivialities, assume that $\mu\in(0,\infty)$. Consider the ...
• 82.1k