# Questions tagged [global-optimization]

The global-optimization tag has no usage guidance.

**-1**

votes

**0**answers

111 views

### KKT for dual of a quadratically constrained linear program

Let $ \mathcal{P}$ be a linear program with quadratic constraints,
$$ \eqalign{
\mathcal{P}: & \min_{x,t} -t \\
& \text{s.t.} \\
{\color{orange}{\mu_k}}: & a^H_k x + a^T_k x^{*} - x^H ...

**8**

votes

**1**answer

490 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

122 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=&\...

**0**

votes

**0**answers

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 ...

**0**

votes

**0**answers

36 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**

votes

**0**answers

60 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

96 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

78 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)^...

**0**

votes

**0**answers

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 ...

**0**

votes

**1**answer

102 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 $1 \leq P \leq N$ ...

**2**

votes

**1**answer

92 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

41 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 ...

**0**

votes

**1**answer

202 views

### Properties of the argmin function (continuity, differentiability..) [closed]

Given vectors $x_1,\ldots,x_N \in \mathbb{R}^d$ and a function, say $\psi \colon \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$, one is interested in the properties of the function $$\Phi\colon (x_1,...

**1**

vote

**2**answers

259 views

### Maximizing a function that is sum of gaussians

Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function
\begin{align}
\mathcal{K}(\mathbf{x},\mathbf{y})=
\alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...

**4**

votes

**2**answers

135 views

### Optimization problem on trace with both the positive semi definite and non positive semidefinite matrix

Given two $N \times N$ symmetric matrices $A, B$, where $A$ is positive semidefinite while $B$ is not positive semidefinite. I am interested in solving unitary constrained trace maximization problem:
...