19 votes
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 ...
18 votes
Accepted

Prove that this expression is greater than 1/2

Let $$f(x,y):=4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Bigl( \sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}(\pi y-\pi x) \Bigr).$$ I will show that $$\min_{0\leq x\leq y\leq 1}f(x,y)=\min_{0\leq x\leq 1/...
  • 86.9k
13 votes

Is all non-convex optimization heuristic?

There has been recently a flurry of new results on provable nonconvex methods which can be guaranteed to converge to the global optimum. In other cases, the non-convex problem itself is shown to have ...
11 votes

"Small" maps from sphere to sphere

Here's an example to show that the infimum is not always attained: Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in ...
11 votes
Accepted

Known configurations maximizing the volume of the convex hull of n points on the unit sphere

The problem is elementary for $n=5$. We may regard that case as a combination of two triangular pyramids sharing a base $\triangle ABC$. Then the volume is always bounded by one-third the area of the ...
10 votes

Generalization of the equilateral triangle?

First, let's assume $a=1$; for other values we can scale a solution with $\sqrt{a}$. So we want to minimize $H=\sum_{i,j} (1-\|x_i-x_j\|^2)^2$. I globally optimized the problem numerically for $n=4$...
9 votes

Prove that this expression is greater than 1/2

Let $F(x,y)$ denote the left-hand side of your inequality. It is easy to see that $|\nabla F(x,y)|\sqrt2/n<0.002$ if $0<x<y<1$, where $n:=6600$. A direct calculation shows that $F(i/n,j/n)&...
6 votes
Accepted

"Most Similar Vector Problem" on an Integer Lattice?

Writing up the comment: You just need to "pixelate" the line by finding all lattice boxes that it crosses: Then the answer vector $v$ must connect to one of the corners of the shaded boxes. Instead ...
  • 4,822
6 votes
Accepted

Software tools for medium-scale systems of polynomial equations

In Maple you can just do with(Optimization): g := (your function): Minimize(g,iterationlimit = 200); On my machine this takes only about 1.5 seconds to return ...
6 votes

Prove that this expression is greater than 1/2

$\newcommand{\R}{\mathbb{R}}$ An advantage of my previous answer was that, while the computer calculations were pretty heavy there, the logic was extremely simple; virtually no thinking or ingenuity ...
5 votes

Solve $\inf_{ X: |X| \le a \text{ a.s.}} E \left[ \frac{1}{1+(X-X^\prime)^2} \right] $

My initial intuition was incorrect. It seems we can sometimes solve this explicitly, and the minimizer does not have full support; my argument is incomplete (as discussed below), but I think it's ...
5 votes
Accepted

A (reverse)-Minkowski type inequality for symmetric sums

Rewrite the inequality in question as \begin{equation*} f(u+v)\le f(u)+f(v) \end{equation*} for $u,v$ in $\mathbb R_+^4$, where \begin{equation*} f(u):=-\left(\left(\frac{1}{\sqrt{u_1}}+\frac{1}{...
5 votes

Does there exist energy-minimizing immersions?

I believe that in general, without any additional assumptions about the manifolds, the answer is in the negative. A counterexample can be found for mappings between annuli. Let $A=A(r,R)$ and $A_*=A(...
5 votes
Accepted

Is this lower bound for a norm of some complex matrices true?

No, it is not; in fact, $2(n-1)$ is a local maximum. Let $B$ be a Hermitian matrix such that $|B_{ij}|=1$ and $B_{ii}=1$. We denote its eigenvalues by $\mu$ (not to confuse them with eigenvalues of $...
5 votes

Maximal distance of $2d+1$ points on a sphere

OK, it's late and I may be wrong but I think that you can obtain the $2d$ points by using any set of orthonormal basis vectors $\{v_1,\ldots,v_d\}$ and their negatives. Now if $n$ is such that a ...
  • 8,851
4 votes

Software tools for medium-scale systems of polynomial equations

To solve a polynomial system, I would try Bertini which is a homotopy-continuation numerical solver that parallelizes extremely well. You can also try to attack the optimization problem directly with ...
  • 1,971
4 votes
Accepted

maximizing a function involving factorial

In a quite large range of the parameters we can approximate the fraction by a Taylor series to obtain $$ f(x) = \frac{1}{x!}\frac{1}{\frac{-\log c}{\binom{x+n-1}{n-1}} + \mathcal{O}\left(\frac{\log^2 ...
4 votes
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
4 votes
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
4 votes
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. ...
4 votes

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
4 votes
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 ...
4 votes

A (reverse)-Minkowski type inequality for symmetric sums

The said claim follows from the following general result on elementary symmetric polynomials, denoted $e_k$ below. $\newcommand{\vx}{\mathbf{x}}\newcommand{\vy}{\mathbf{y}}$ Theorem A (S. 2018). $\,...
  • 27.9k
4 votes

Maximal distance of $2d+1$ points on a sphere

Turns out kodlu's idea works in all dimensions, regardless of the existence of any Hadamard matrices. Consider all coordinate permutations of $$(1,...,1,-d)\in\Bbb R^{d+1}\quad\text{and}\quad (-1,......
  • 10.6k
4 votes
Accepted

Measurable selection for argmin to distance

Here is a simple direct argument. For $i=1,\ldots,n$, let $$C_i=\{z\in Z\mid d(z,x_i)\leq d(z,x_j), j=1,\ldots,n\}.$$ Clearly, each $C_i$ is closed and hence measurable. Let $M_i=C_i\setminus\bigcup_{...
3 votes
Accepted

Bounding the difference in the value of a strongly convex function at its integer minimum and other integer points

Unfortunately, no. Here is an example for $n=1$ (1-dimension). For parameters $m>0$, $b\in\mathbb{R}$ define: $$f(x) = (m/2)(x-b)^2 $$ For any $b \in \mathbb{R}$, this function $f$ is strongly ...
  • 524
3 votes

Two fold optimization: is there an established approach for this kind of problem?

If the coefficients of your linear expressions are integers and you want the variables $x_i$ to be integers as well then the problem can be written as an integer program: \begin{align*} \text{Maximize ...
3 votes

iterative solution better than analytic solution?

It may very well be that the evaluation of the analytical solution you are referring to poses some problems. An elementary example is an analytical formula for the solution of $Ax=y$ where $A$ is an ...
  • 7,824
3 votes

Software tools for medium-scale systems of polynomial equations

As far as I can see, you'd like to find a global minimum of $F_0(x)=\sum_{k=1}^{M_0} f_k(x)^2$, where $x=(x_1,\dots,x_{12})$. Equivalently, the problem is to find maximal $\mu$ so that $F(\mu,x):=F_0(...
3 votes

Maximizing a pseudoconcave function in a box

Your problem is a special case of a Fractional Linear Program, so as such following the recipe provided on Wikipedia you should be able to solve it by using a reformulation to an equivalent linear ...
  • 27.9k

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