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Questions tagged [optimization]

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

22512 questions
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Find the maximal value of $\mu^{-1}\cos\theta+\sin\theta$

This is part of a physics problem I was doing yesterday. I am supposed to find the maximum value of $$\mu^{-1}\cos\theta+\sin\theta$$ This is is supposed to produce the result of $\sqrt{1+\mu^{-2}}$. However I am not getting that. By taking the…
Physics_Learner
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Optimization with implicit constraints

I want to minimize the following objective function: $\min_{w,v} f(w,v) + g(w,a^*) \qquad (*)$ where $a^* = \arg\min_a h(w,a) $. If I can have an analytical solution for $a^*$ with respect to $w$, then I can put it back to…
user3138073
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Minimizing the function $\sqrt{2x} - \left\lceil \frac{\sqrt{1+8x}}{2} \right\rceil$ for $x > 0$

How can you find the minimum of $\sqrt{2x} - \left\lceil \frac{\sqrt{1+8x}}{2} \right\rceil$ for positive integer values of $x$?
user35671
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Single maximum quasiconcave

Let $f(x)$ a positive decreasing real function with a positive domain. If $(1/f)$ is convex, is it true that $h(x)=xf(x)$ has a unique maximum?
Quema
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Optimization Pb

Let $\lambda_1, \lambda_2 \geq 0$ and consider the problem : $$\begin{array}{ll} \text{minimize} & x^2(1-2\lambda_2)+\lambda_1x+3y^3+\lambda_1y\\ \text{subject to} & 2x+y=2\\ & x,y \geq 0\end{array}$$ I saw that the curve $~2x+y=2~$ with $x,y \geq…
Sylvain Lhermite
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Optimization over a partition

Let $u(x):R\rightarrow{}R$ a real function. And a vector of integer numbres $\mathbf x =(x_1,x_2,...x_n)$ with $x_1\geq{}x_2\geq{}...\geq{}x_i\geq{}0\geq x_{i+1}\geq...\geq{}x_n.$ Let $S=\left\{{\mathbf y_1, \mathbf y_2, ..., \mathbf y_t}\right\}$…
Quema
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Path integral - Feynman integral

I have a question regarding Physics. The theme concerns some history behind the development of the physics we know and the questions it might evoke. In quantum physics, there is a notorious effort to assemble its different parts: gravitation,…
Bruno Lobo
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Optimisation when state evaluation is costly.

Looking for advice on constrained optimization problem. Currently I construct a Lagrangian and use various descent algorithms, line searches, all the usual tricks. No matter what I use, I am limited by a very costly state-evaluation step. Is there…
AngusTheMan
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New store branch locations optimisation problem

Imagine we manage a chain of stores. We already have a number $M$ of branches of this chain in a certain region. The CEO has decided we need to expand and should open $N$ new stores in this same region. The question is: Where should we place these…
Roberto
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Non convex optimization

For $u$ a unit vector and $x_1\ldots,x_n$ a set of fixed points in $R^d$, consider $f(u)=\sum_{i=1}^n \frac{\langle x_i,u\rangle_+}{\langle x_i,u\rangle_-}$, where $a_+$ and $a_-$ are the corresponding positive and negative parts of a real $a$. Is…
Mike_Math
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Optimization variation of Hellys theorem

I'm trying to solve the following variation of the Hellys theorem problem: Let $B = {B(o_i,r):i = 1,...,n}$ a set of balls with radius r with center at $o_i\in\mathbb{R}^2$. Compute the minimal radius r such that: $\bigcap_{i=0}^n…
AvivSham
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Backtracking Line Search Algorithm - Why make $a$ smaller every time?

I am studying line search methods and I stumbled upon the "backtracking algorithm" for calculating a step length $a_k$ that gives a sufficient decrease in our function. To actually compute this sufficient decrease, we use the first Wolfe condition,…
Thomas
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exists r > 0 such that ∇2f (x) > 0 for any x ∈ B(x0,r).

Let f : U ⊆ Rn → R be a twice continuously differentiable function. Let x0 be an interior point of U such that ∇^2f(x0)> 0(hessian). Prove that there exists r > 0 such that ∇2f (x) > 0 for any x ∈ B(x0,r). i tried to prove it with minor criteria .
Noam Ben Shimol
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Minimizing $f(x)=x^{4/3}$ using Newton's method and show as long as initial approximation is not $0$, the algorithm does not converge to $0$.

Consider the problem of minimizing $f : \mathbb{R} \to \mathbb{R}$ be given by $f(x) = x^{4/3}$ . Note that $0$ is the global minimizer of $f$ . (a) Write down the iterations for Newton’s method for this problem. (b) Show that, as long as the…
mathnerd
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Stockpile water usage optimization

I have been asked to model a Calcium Sulfate dihydrate stockpile (A cone shape with an angle of repose), that is washed with water to remove impurities of MgCl2 and KCl. My job is to determine how much water should be used when given a stockpile's…
James
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