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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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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44 views

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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54 views

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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Computing maximal value of a function with constraints using software - how to make the computation rigorous?

I have a fairly messy rational function $D$ in variables $u_1,u_2,u_3,q$. I am trying to compute the maximum of this function on the compact set $K$ defined by $u_1^2+u_2^2+u_3^2=1, 0 \leq q \leq e^{-...
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22 views

Global extreme points - Find the biggest value

I have the following function and constraints: f(x,y,z)=(x-1)^2+(y-1)^2+(z-1)^2 Domain: x^2+y^2+z^2≤2 and z≤1 I have already excluded stationary points so I want to use Lagrange multipliers ...
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35 views

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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Numerically solve $\inf_w\sup_{1\le j\le n}\sum_{i=1}^mw_ia_{ij}$ subject to $\sum_{i=1}^mw_i=1$

Let $m,n\in\mathbb N$, $$W:=\left\{w\in[0,\infty)^m:\sum_{i=1}^mw_i=1\right\}$$ and $a_{ij}\ge0$ for $i\in\{1,\ldots,m\}$ and $j\in\{1,\ldots,n\}$. How can we solve the saddle-point problem $$\inf_{...
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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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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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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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67 views

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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64 views

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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20 views

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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235 views

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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71 views

$2$-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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1answer
109 views

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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451 views

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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186 views

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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125 views

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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109 views

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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558 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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513 views

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 ...