Mathematics

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.

56 questions
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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 looking for the minimizing configuration of the function…
user268272
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Global Optimization and Real Algebraic Geometry

Wikipedia suggests that: " Methods based on real algebraic geometry " are some of the " most successful general strategies " for solving global optimization problems. Could someone suggest an reference for learning about how algebraic geometry can…
Nathaniel Bubis
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Is the minimum of this optimization problem essentially unique?

Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$ , and suppose that $h(x)$ is strictly increasing on $[1,\infty)$ , and strictly decreasing on $(0,1]$ . Let $s>0$ be a parameter, and define $…
Asaf Shachar
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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 with restrictions like Banach-spaces or…
welahi
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2 answers

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 strategy: Use of interval arithmetic in order to be…
Mathieu Dutour Sikiric
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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 large minimum. If you zoom out, (e.g. when the x…
yannick
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Is this function convex and does it have a global minimum for $ x > 0 $?

I'm studying the following function, defined for $ x > 0 $ , where all constants $ c_1, c_2, c_3, c_4, c_5 $ are positive real numbers: $$ f(x) = \frac{c_1}{x} + \frac{c_2}{\log_2\left(1 + \frac{c_3}{c_4 x + c_5} \right)} $$ I would like to…
Osama Alshalali
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Help in solving an optimization problem

I have a simple optimization problem; I already know the answer but I don't know how to prove it! Assuming we have a series of positive real numbers, $$ x_1, x_2, x_3, ..., x_n, $$ we want to find a series $$ y_1, y_2, y_3, ..., y_n $$ that…
Amir Hashemi
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Global optimization of problem with L1 regularization

UPDATED : I was forgetting this basic fact (that the sum of convex functions are themselves convex), and therefore I thought that I needed a global optimizer. I now see that that is not the case and my problem does indeed reduce to the classical…
tbolind
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Minimizing the real part of $e^{-ik\varphi} \left(1-(1-e^{i\varphi})\frac{k}{n}\right)^n$

Let $n \geq 3$ and consider the family of functions $\{f_k\}$ indexed by $k = 1,2,\ldots,n-1$ , defined on the unit circle via $$f_k(\varphi) = \textrm{Re}\left[e^{-ik\varphi}\left(1-(1-e^{i\varphi})\frac{k}{n}\right)^n\right] = \sum_{j=0}^n…
meler
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Escape from local minima in the gradient descent method

I'm using gradient descent $$x_i=x_{i-1}-\gamma\nabla f(x_{i-1})\tag1$$ to minimize a function $f$ . Now, I've observed that I got stuck in a local minimum when $x_0$ was chosen in an unlucky way. Is there any mechanism to detect that we've got…
0xbadf00d
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What is the name for this type of optimization problem?

As we all know, a classic optimization problem can be represented in the following way: Given: a function $f: A \rightarrow \mathbb{R}$ from some set $A$ to the real numbers Sought: an element $x_0 ∈ A$ such that $f(x_0) \le f(x)$ for all $x \in A$…
Shaun Han
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Show that there is no global minimum

I want to show that the multivariablefunction $$f(x,y)=2x^4+5y^4-|x|-\sqrt{|x|+|y|}$$ has no global minimum. For that do we calculate the critical points to get the desired result? Or do we suppose that there is a global minimum at a point $(a,b)$…
Mary Star
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Global minimum of convex function

Let $E \subset \mathbb R^n$ a closed set, convex and not bounded. Let $f \in C^0 (E, \mathbb R)$ strictly convex, it means that $$\forall (x, y) \in E^2, \forall \alpha \in ]0, 1[, x \neq y \implies f(\alpha x + (1 - \alpha)y) < \alpha f(x) + (1 −…
Ice Tea
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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 maximum in a function and remembered that…
Marcus
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