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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Texts about global minima in functionals

I am doing some numerics where I found the minimal value of a functional that does not satisfy the Euler-Lagrange equation associated. I think I am dealing with a minimal value that is not a local ...
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Extended and well detailed bibliography for the cubic algorithm in global optimization of Lipschitz-continuous functions.

Does anyone know which could be a nice bibliography for the topic? I'm headed to develope the cubic algorithm (maths and code) for global optimization of Lipschitz-continous functions with several ...
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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 ∈ ...
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Global optimization bibliography [closed]

Can anyone recommend some books about global optimization? I will have to make a work about that topic despite I don't know exactly the title yet, so any level and approach is welcomed. Update [Jan ...
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Number Of Global Optima in Single Solution Metaheuristics

I am reading the book "Metaheuristics From Design to Implementation" written by El-Ghazali Talbi and on page 91, in the "Single-solution based metaheuristics" section, he says ...
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Where can Global Maxima and Minima occur in a Bounded Region in multivariable calculus

I was watching https://www.youtube.com/watch?v=Hg38kfK5w4E&ab_channel=TheOrganicChemistryTutor about finding global max and min for a multivariable function. He said that they occur at at the ...
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Extreme value of a multivariable function under constraint

I have two functions of $\{a,b,c\}$, $$ W=\sqrt{2 (\cos (a)-\cos (b)-\cos (c))+3}+\sqrt{2 (-\cos (a)+\cos (b)-\cos (c))+3}+\sqrt{2 (-\cos (a)-\cos (b)+\cos (c))+3}+\sqrt{2 (\cos (a)+\cos (b)+\cos (c))+...
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Relationship between local minimum and global minimum based on metrics

Given a set of points $S \subset X$ with a metric defined $d : X \times X \to \mathbb{R}^+$. I am interested in the relationship between $Z = \textbf{argmin}_{z \in X} \sum_{i \in S} d(i, z)$ and $S$....
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Difference between global lower bound and tight lower bound for minimization problem

As for minimization problem, What is the difference between global lower bound and tight lower bound ( which is larger than the global one) with respect to the quality of solution in case of ...
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Does Globally convergent homotopy guarantee optimality for non-convex problems?

I just came across this algorithm in an engineering paper in which I believe the author claims It guarantees global optimality in non-linear systems but never mentioned the convexity of the problem. ...
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Approximately-Global Nonlinear Multivariate High-Dimensional Optimization of *Differentiable* Nonconvex Scalar Function

Title says it all. What methods do you recommend to solve this specific problem? I can build most solutions you may suggest in Python, where I'd like the solver to be. I've spent hours and hours ...
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KKT points for minimization problem?

I have the following problem: $$\begin{array}{ll} \text{minimize} & \frac{1}{2} \|x \|_4^4\\ \text{subject to} & \| x \|_2^2 - 1 = 0\end{array}$$ and I don't know how to start. Maybe I can ...
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2 answers
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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 ...
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Show that the maximumof $f$ isglobal [closed]

I want to show that $f(x,y)=xye^{-x-y}$ has a global maximum in $(1,1)$. So we have to show that $f(x,y)\leq f(1,1)$, or not? We have that $$xye^{-x-y}=\frac{xy}{e^{x+y}}\leq xy$$ But how can we ...
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multicriteria optimization with two variables : what criteria to choose and what optimization technique?

I want to optimize two parameters, let's call them $x$ and $y$, in a bounded domain, that describe a system by setting a function $\beta$'s behaviour. What I look for, basically, is a $\beta$ function ...
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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(...
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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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Finding global maximum of a 2-variable function

Let $$f(x,y) = -e^{x^4+y^2}+x^2+y^2+1$$ Find global/local maximum and minimum and upper/lower bound of the function. I found that the lower bound is ${-\infty} $ and also that, since $$-e^{x^4+y^2}\le ...
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How to show that $\left\{x^*\in X \mid \limsup_{y \to x,\;\|y\|=1}\frac{\langle x^*,y-x\rangle}{\|y-x\|} \le 0\right\} = \mathbb R x$ if $\|x\|=1$

Let $X$ be a Banach space with topological dual $X^\star$. Given an extended value function $f:X \to \mathbb R \cup \{+\infty\}$, defined its Fréchet subdifferential at a point $x \in X$ as $$ \...
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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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Linear Objective Minimization on Intersection of Two Ellipsoid Surfaces

Let $D=\text{diag}(d_1,d_2,\dots,d_n)$ be a positive definite $n\times n$ matrix, $0\ne c\in \mathbb R^n,$ and $\alpha$ be a positive real number such that $\alpha \ne d_i$ for $i=1,2,\dots,n$. ...
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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&...
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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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What if we have 2 same local extrema, how do we deal with global extrema?

I wonder, if I have to analyze some function and find global extrema. for example this function: It has 2 equal local maxima. So what local maximum should I settle as global? Can I both or have to ...
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Find a global maximum and show that it is unique

Let $a_{i} \in \mathbb R_{>0}$ and consider the following problem: $$\begin{array}{ll} \text{maximize} & f(x) := \displaystyle\prod_{i=1}^{n} x_{i}^{a_{i}}\\ \text{subject to} & x \in S\...
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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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Can we find a minimizer of $\sum_{i=1}^k\int a_iw_i$ satisfying $\sum_{i=1}^kw_i=1$?

Let $(E,\mathcal E,\lambda)$ be a measure space; $f:E\to[0,\infty)^3$ be $\mathcal E$-measurable with $\|f\|\in\mathcal L^2(\lambda)$; $\tilde p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ for some $\...
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Can we find a minimizer of $\sum_{i=1}^ka_iw_i$ satisfying $\sum_{i=1}^kw_i=1$?

I've got a quite simple problem: Given $k\in\mathbb N$ and $a_1,\ldots,a_k\ge0$, I want to find $w_1,\ldots,w_k\ge0$ with $\sum_{i=1}^kw_i=1$ minimizing $\sum_{i=1}^ka_iw_i$. Are we able to solve ...
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2 votes
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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 ...
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how to find global minimum and max over set s

I need to find the global minimum and maximum points of the linear function $f (x, y) = 5x − 8y$ over the set $S =\{(x, y) \in \mathbb R^2: 5x^2 − 8xy + 4y^2 + 8x − 8y ≤ 5\}$
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$f(\theta)<f(\theta^*)$ $\forall \theta\in \Theta$ implies $\theta^*=argmax_{\theta\in \Theta}f(\theta)$

Take $\theta^*\in \Theta\subseteq \mathbb{R}^K$ and let $f$ be a real function of $\theta$. If $f(\theta)<f(\theta^*)$ $\forall \theta\in \Theta$, with $\theta\neq \theta^*$, then does the ...
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Do initial parameters in this global optimization problem matter?

Let's say I have some data to adjust to this model: $$V=V_0 +K_1(\cos (\omega_1 +f_1)+e_1\cos(\omega_1)) + K_2(\cos (\omega_2 +f_2)+e_2\cos(\omega_2)), $$ with $V_0$, $K_1$, $K_2$, $\omega_1$, $\...
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5 votes
2 answers
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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 votes
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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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1 vote
1 answer
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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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4 votes
2 answers
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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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1 vote
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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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0 votes
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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 ...
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15 votes
2 answers
639 views

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 ...
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Global min-max optimization

When is \begin{equation} \min_X \max_Y f(X,Y) \end{equation} globally solvable? I.e., when can we find global solution for the optimization problem? I am not looking for reformulations. Is it only ...
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