# Questions tagged [global-optimization]

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### Properties of $l_q$-balls

For a given $q\in (0,1]$, define the $l_q$-ball as $$\mathbb{B}_q(R_q)\mathrel{:=}\left\{\theta\in\mathbb{R}^d\,\middle\vert\,\sum_{j=1}^d \lvert\theta_j\rvert^q\leq R_q \right\}.$$ For a given ...
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### How do i get the $\Delta_a$ value that would maximize $(\Delta_d-\Delta_a)$

so this problem is derived from a real world problem i have so i'm unsure if it can be solved through maths or not. I'm currently "finding" the solution through tons of computer iteration ...
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### A question about strong slopes (nonsmooth analysis)

Context. I'm reading the manuscrip "Nonlinear Error Bounds via a Change of Function" by Dominique Azé and Jean-Noël Corvellec (J Optim Theory Appl 2016), and I'm having a hard time ...
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### Pros and cons of using integer programming alone or combined integer and global optimization?

First, I am not sure if this is the right question to ask in this forum. But I have been looking for answers for a long time and I have been also asking my university's "engineering" professors but I ...
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### When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For ...
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### Joint optimization of the order 1 moment of a function and its Fourier transform

For the purpose of a quantum optics experiment, I come to the following problem : Let $X,P \in \mathbf{R}^2$ \begin{equation} J(\psi) = | \int_{-X}^{X} x |\psi^2(x) | dx + \int_{-P}^{P} p |Tf(\...
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### Is there a multiplier rule for this minimization problem?

Let $(E,\mathcal E)$ be a measurable space, $W\subseteq\left\{w:E\to\mathbb R\mid w\text{ is }\mathcal E\text{-measurable}\right\}$ be a Banach space, $k\in\mathbb N$ and $f:W^k\to[0,\infty)$. I'm ...
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### Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related?

I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy: $$\inf_{x\in\mathbb{R}^n}\quad p(x),$$ where $p$ is a polynomial of even degree ...
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### Fritz-John conditions: Equality-constrained case as special case of inequality constraints

In Chapter 4 of Nonlinear Programming: Theory and Algorithms by Bazarra, Sherali, and Shetty, the following claim is made after Theorem 4.3.2 (Fritz-John necessary conditions): "Note also that these ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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,...
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### 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\...
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: ...