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.
Questions tagged [global-optimization]
55 questions
25
votes
3 answers
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…
user268272
16
votes
2 answers
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 be…
Nathaniel Bubis
- 33,425
6
votes
1 answer
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 $
F(s)=\min_{xy=s,x,y>0}…
Asaf Shachar
- 25,967
5
votes
1 answer
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
- 283
5
votes
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…
4
votes
3 answers
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
- 277
2
votes
1 answer
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
- 21
2
votes
0 answers
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
- 45
2
votes
0 answers
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 $$f_k(\varphi) = \textrm{Re}\left[e^{-ik\varphi}\left(1-(1-e^{i\varphi})\frac{k}{n}\right)^n\right] = \sum_{j=0}^n…
meler
- 185
2
votes
2 answers
Escape from local minima in the gradient descent method
I'm using gradient descent $$x_i=x_{i-1}-\gamma\nabla f(x_{i-1})\tag1$$ to minimize a function $f$. Now, I've observed that I got stuck in a local minimum when $x_0$ was chosen in an unlucky way.
Is there any mechanism to detect that we've got stuck…
0xbadf00d
- 14,208
2
votes
0 answers
What is the name for this type of optimization problem?
As we all know, a classic optimization problem can be represented in the following way:
Given: a function $f: A \rightarrow \mathbb{R}$ from some set $A$ to the real numbers
Sought: an element $x_0 ∈ A$ such that $f(x_0) \le f(x)$ for all $x \in A$…
Shaun Han
- 107
2
votes
2 answers
Show that there is no global minimum
I want to show that the multivariablefunction $$f(x,y)=2x^4+5y^4-|x|-\sqrt{|x|+|y|}$$ has no global minimum.
For that do we calculate the critical points to get the desired result?
Or do we suppose that there is a global minimum at a point $(a,b)$…
Mary Star
- 14,186
- 15
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- 205
2
votes
1 answer
Global minimum of convex function
Let $E \subset \mathbb R^n$ a closed set, convex and not bounded. Let $f \in C^0
(E, \mathbb R)$ strictly convex, it means that $$\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
- 41
2
votes
1 answer
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
- 213
2
votes
0 answers
Proof KKT points are inside a ball and find the Lagrange multipliers
I am trying to learn continuous optimization and I need to solve the following exercise. Despite the fact that I solved some exercises about KKT conditions and Lagrange multipliers, I can't solve this one. It seems to require an upper level of rigor…
Pedro Delfino
- 327