# Questions tagged [global-optimization]

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156
questions

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54 views

### How can we analytically solve this max-sum-min problem?

Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...

**1**

vote

**1**answer

73 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 ...

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41 views

### Optimizing Function over Configuration Space

Below, I describe a function defined on (a certain subset of) the configuration space of $n$ points on the plane and I want to know how its maximum grows as $n\to\infty$. Before describing the ...

**6**

votes

**4**answers

2k views

### Prove that this expression is greater than 1/2

Let $0<x < y < 1$ be given. Prove
$$4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Big[
\sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}[\pi(y-x)] \Big] \geq \frac{1}{2}$$
I have been working on this ...

**10**

votes

**1**answer

520 views

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

I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been ...

**5**

votes

**1**answer

130 views

### An effective way for the minimization of $\left\|ABA^{-1}-C\right\|$

Supposing I have complex square matrices $B_i$ and $C_i$ ($i = 1,\dots,N$) of dimension $4 \times 4$.
Is there an effective algorithm for solving the following problem?
$$\begin{align}
A=&\...

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19 views

### Can the partial concavity of the following decomposable objective function be used for optimization?

The problem I am trying to solve is the following:
$$\begin{array}{ll}
\min & f(x)+g(y) \\
\mathrm{s.t.} & y\ge x\ge 0,\\
\ & p\le ax+by\le q,
\end{array}$$
where $a,b,p,q$ are ...

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37 views

### Stochastic process for minimize a mean

I have the next problem: consider an inventory process $\{X_{k}\}$ such that $X_{k+1}=X_k+A_k-\xi$, $X_0=25$, where $A_k$ is the number of items of items produced at the $k$th-month and $\xi$ is the ...

**5**

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61 views

### On a maximum of a determinant with dependent variables

Let $x_1,\ldots,x_n\in [-1,1]^n$ and define the function
$$f(x_1,\ldots,x_n):= \prod_{i=1}^n\prod_{j=i}^n\left(1-\prod_{k=i}^j x_k\right).$$
This is a positive function, and actually coincides with ...

**1**

vote

**1**answer

104 views

### Minimizing sum of ratio of linear functions (Sum of Linear Ratios Problem)

Given constants $c_i \in \mathbb{R}$ and $d_i \in \mathbb{R}$ and variables $x_i \in \mathbb{R}$, where $c_i > 0, d_i > 0, x_i > 0$ can we easily solve the following optimization problem:
$$...

**1**

vote

**0**answers

81 views

### When are quadratic integer programs “easy to solve”?

Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to maximize $f$ on $N$. $f$ has the following form
$$
f(n) = \sum_i A_i n_i -\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^...

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votes

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32 views

### symmetry preservation and symmetry breaking in optimization algorithms

Let $M$ be a Riemannian manifold, and $G$ be a group of isometries on $M$. If we have a $G$-invariant functional $F: M \rightarrow \mathbb{R}$, i.e. $F(g \cdot x) = F(x)$ for all $x \in M$ and $g \in ...

**1**

vote

**1**answer

114 views

### Finding $P$ points among $N$ to approximate a probability density function?

Let $f$ be a probability density function (positive such that $\int_{\mathbb{R}} f(x) \mathrm{d} x = 1$) and $X_0 = \{x_n\}_{1\leq n \leq N}$ be $N$ given real points. We also fix $1 \leq P \leq N$ ...

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**2**answers

128 views

### Looking for a very particular kind of non-convex functions

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously,
It should be at least twice differentiable.
It should have a ...

**1**

vote

**0**answers

45 views

### Minimum Preserving Transformations [closed]

If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then
$$
\operatorname{argmin}_{x \in X} f(x)
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$
X and Y ...