# Questions tagged [global-optimization]

Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set.

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### Question on continuity and optimization

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a continuous function on $\mathbb{R}^2$. Is it true that, if $$\lim_{||(x,y)||\to{\infty}}{f(x,y)}=-{\infty}$$ then $f$ has a global maximum on $\mathbb{R}^2$...
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### Proof global minimum in two dimensions WITHOUT hessian but based on limit behavior

Consider a function $\Phi \colon D \to \mathbb{R}$, $D = \{(x,y) \in (0,\infty)^2 \mid x < y\}$, $\Phi$ differentiable on $D$. It is known, that $(x^\ast,y^\ast)$ is the only critical point in $D$ ...
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### Coercive/(weakly) semicontinuous function: extreme values

Consider functionals of the form $$\phi : X \rightarrow \mathbb{R} \cup\{+\infty\},$$ where $X$ is an arbitrary, normed vector space. In particular, $X$ may be of infinite dimension. I would be fine ...
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### Why is it difficult to find the Global Optimum?

When studying Calculus I learnt that it it possible to take the derivative of a function to find its minimum and maximum points. I then wondered what happens if there are more than one minimum and ...
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### Is there any way to solve df(t)/dt = 0 for t and f(tsolved) value if we know F(s), the Laplace transform of f?

I am trying to know max(f(t)) value but I have only F(s) equation of it and I thought that by solving df(t)/dt == 0 by Laplace using F(s) which is s*F(s) == 0, I can easily solve what is t for max(f(t)...
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### Finding $n$ points $x_i$ in the plane, with $\sum_{i=1}^n \vert x_i \vert^2=1$, minimizing $\sum_{i\neq j}^n \frac{1}{\sqrt{\vert x_i-x_j \vert}}$

Let $x_1,..,x_n$ be points in $\mathbb R^2$ under the constraint $$\sum_{i=1}^n \vert x_i \vert^2=1.$$ So not all the points are on the circle, but their sum of the norms is constrained. I am ...
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### Rigorous global optimization

The work by Thomas Hales (see enter link description here) before the formal proof solves a number of global optimization problems that need to be solved exactly. The strategy relies on following ...
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### Is there any algorithm to find the global minimum for the quasi-convex optimization?

Consider the following optimization problem $$\begin{array}{ll} \text{minimize} & f(x)\\ \text{subject to} & g(x,y) \leq 0\end{array}$$ where the objective function $f$ is linear. When I fix ...
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### Any comprehensive books on global smooth optimization?

Can you share any information about books that review most of the existing numerical methods for global minimization of a multivariable objective function? The objective and constraints are assumed ...
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### How to find potential near-optimum clusters after random sampling?

I want to run local optimizers on a multi-dimensional function $f$ with several local minima. To make sure to trap into the real global optimum, I am running first an initial sampling (e.g. randomly)...
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### Can basin-hopping be applied for global constrained optimization?

Can we implement Basin-Hopping approach for constrained optimization problems ? The literature suggests it for unconstrained global optimization, but I have a few constraints. Can it be adapted for ...
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### $2$-variable optimization problem — global maximum

In the last couple of weeks, I have been dealing with a two-variable optimization problem, which I have been unable to solve. In the problem given below, I am attempting to show that the unique local ...
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### Guarantee of global minimum for a given optimization problem.

I've been self-studying my way through A First Course in Probability by Ross, and just finished working through an example dealing with optimization of a function of a continuous random variable. The ...
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### Global optimization of non-smooth function

I have a number of functions (see for example two of them down below), and I need to find their global optimum for each of them. They are non-smooth, but they are always funnel-shaped, exhibiting a ...
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### Global optimization of a well-defined function with gradient information

I try to minimize the function $$f(x_1, …x_n)=\sum\limits_{i}^n-a_icos(4(x_i-b_i)) +\sum\limits_{ij}^{edge}- cos(4(x_i-x_j))$$ $$x_i,b_i\in (-\pi, \pi)$$ where $$\sum\limits_{ij}^{edge}$$ only ...
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### Global Optimization, symmetric solutions

Does anyone have the idea to solve the global multivariate minimization problem as below? \text{minimizes}\quad (x_1x_2x_3+x_1x_4x_5+x_1x_6x_7+x_2x_4x_6+x_2x_5x_7+x_3x_4x_7)-(x_1+x_2+x_4+x_7) \\ \...
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### Global optimization

Assume that I want to find the global minimum of a non-linear, non-convex, multidimensional function subject to several restrictions. Could you recommend me any deterministic strategy which can ...