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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").

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$\arg\min_\theta [f(\theta x)+f(\theta y)]=\arg\min_\theta [\log f(\theta x)+\log f(\theta y)]$ for all $x,y$ and monotonic $f$?

I would like to know whether the following statement is true for all $x,y \in \mathcal{R}$: $\arg\min_\theta [f(\theta x)+f(\theta y)]=\arg\min_\theta [\log f(\theta x)+\log f(\theta y)]$ where $f$ is any monotonically increasing function.
user2170028
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Minimum of sum of square roots in matrix notation

I'm trying to find an efficient approach to minimising the sum of pseudo-huber penalty along a vector. Ideally I would like to be able to formulate this problem as a linear system of the form $\mathbf{Ax = b}$, which can be solved via the conjugate…
Stephen Tierney
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Optimization problem: Finding the maximum value

Can someone please give me a hint on this problem? I want to find the maximum value of y, given the equations:
Niousha
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maximize $x^a + y^b$ subject to $p_1x+p_2y=w$ utility max.

This is a utility maximzation problem maximize $x^a + y^b$ subject to $p_1x+p_2y=w$ (utility maximization problem) Anyone has any idea, there are no restrictions on $a$ and $b$, as far as i can see it. many thanks!!!
Alex
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Optimization Problem

Can someone please help me with this minimization problem? I dunno what to do after replacing p(x) with given s.t.
Shady
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Perimeter equation

A window is in the form of a semicircle surmounted over a rectangle. Thew total perimeter of the window is 12m. Note:This is a part of a maxima minima question that I was trying to solve. I could solve it but my answer was different from other…
Rajath Radhakrishnan
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Find Maximum of Function $L(p)=\prod_{i=1}^{20}\left[ (1-p)A_{i}+pB_{i}\right]$

The $A_{i}'s$ and $B_{i}'s$ are known. I seek the $p$ which maximizes $L(p)$. I thought it might be easier to maximize $\log L(p)$ instead $L(p)$, but I think it is a dead end $\log L(p)=\sum_{i=1}^{20}\log \left( (1-p)A_{i}+p B_{i}…
Mael
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Regulating the speed of a clock

I want to regulate the speed of a clock but with each regulation there is a random change whose size increases with the size of the intended change. Formally speaking, let $a_n$ be the error in the speed of the clock after $n$ corrections. Knowing…
finkelsteinabcd
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4 convex sets in a plane have a point in common

Let $X_1,X_2,X_3,X_4$ be four sets in the plane such that any three of them have a point in common. Do all four of them have to have a point in common? What if sets are convex? Attempt: I think all of them should not have a point in common, but I…
Koba
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Minimizing an objective with sign functions?

Assume $a\in {{R}^{d}}$ and $B\in {{R}^{d\times d}}$. Consider minimizing an objective function of the form $\text{sgn}{{\left( a \right)}^{T}}B~\text{sgn}(a)$ wrt to $a$. Assume it is bounded below. Is replacing the sign functions with…
Adam I.
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Minimization problem with PMP

Problem: Solve $\int_0^1(x^2+u^2)dt \rightarrow min$, subject to $x'(t) = u(t) + x(t)$ and $x(0) = 0$. Our approach to the solution We solve this problem using the Pontryagin Maximum Principle. We know that $x_1$ is free so we can use the…
Nedellyzer
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Maximizing Value of Function

I have three variables $p \geq 0$, $q \geq 0$, $r \geq 0$ and a positive constant $m$. Let $m = p + q +r$. How can I show that the maximum value of $pq + r$ is no more than $\frac{m^2}{4}$? It's easy to see that decreasing $r$ (thus increasing $p +…
Arimelho
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Relaxed optimization problems

The original problem is \begin{align} \min & f(x) \tag{1}\\ \text{s.t.} & \text{constraint 1} \tag{2}\\ & \text{constraint 2} \tag{3}\\ \end{align} However, it is very hard to deal with constraint 2. Therefore, I just solve the objective function…
MIMIGA
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Optimization with definite integral

if I have to maximize $\left[\int_0^N c(i)^k\,\mathrm di\right]^{1/k}$ subject to $\int_0^N p(i) c(i) \,\mathrm di \leq I$ where $I, k$ are constants, and $c(i)$ is our choice variable. I saw in the solution that the FOC of this problem is: $k…
flavis
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Formulating non negative least square problem

I was reading the paper “Fitting a graph to vector data [pdf]” and I found this optimization problem: $$ \min_{w,s} ||Mw||^2 + \mu||\mathbf{1} - Aw - s|| $$ subject to $w,s\geq 0$, where $M$ is a $d\cdot n\times m$ matrix, $A$ a $n \times m$ one and…
daureg
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