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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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Is there anyone who can give a hint for this Qn

$x^* \in R^3$ Maximize: $f(x)$ Subject to: $g_1(x) \le 1$ and $g_2(x) \le 3$ $g_1(x^*) = 1, g_2(x^*) = 2, ∇f(x^*) =4∇g_1(x^*)$ These are given If first constraint change to $g_1(x) \le 0.99$, but second stays same. Approximately what is the maximum…
Jumbo09
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Optimization Question Finding closest point

Find the point on the graph of the function that is closest to the given point. The function: $f(x)= \sqrt {x-8}$; the point: $(15,0)$
Haris Rahim
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Changing model function in non linear curve fitting

I have some $(t,S)$ data and this four-parameters stretched exponential model (from diffusion NMR signals): $S(t) = S_0\cdot e^{-D\cdot t^a}+c$ . Here $D$, $S_0$, $a$, $c$, are parameters to estimate. They must be all positive except for $c$ that…
Mik2A
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Optimization works but: different cost when using LM compared to all others

I am quite new to the field of optimization, but have one question: I am using python and scipy to optimize my objective function (quite complex so I dont want to write it in here). I know the analytical gradient and hessian, so all optimizers can…
horsti
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Additional assumptions for the robust optimization paradigm

Please, see here at page 3: could you help to understand the proofs from E.1 to E.4? In particular: why, by adding the constraint $\operatorname{c^Tx}-t\leq 0$, did the objective function become certain? What's the meaning of uncertain right-side…
Marco Pittella
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Difficulty finding Dual

min $\left \langle a,x \right \rangle$ subject to $\left \| x \right \|^2 \le 1$ What was done: $L(x,\lambda) = a^Tx + \lambda^T(x^Tx-1) = a^Tx + \lambda^Tx^Tx-\lambda^T$ that is, $L(x,\lambda) = inf\{(a^T+\lambda^Tx^T)x\}-\lambda^T$ Gradients…
Amissadai ferreira
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How to formulate weighted distance?

Given we have 2 trees and their coordinates on xy plane is : $(-2,1)$, $(0,2)$ and their weights are $3$ and $4$ respectively. We want to plant a new tree with the condition that the weighted sum of the absolute distances from the existing trees…
moli
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A single-machine optimization challenge

Can someone point me in the right direction to solve the following? I have a machine that is able to process a batch containing any number of items, each with some length $L_n$, as long as the total length of all items in each batch does not exceed…
mdho
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Model a given problem using Multiobjective and Goal programming.

An electronics manufacturing company sets up security alarms for the market ersidential security. It produces three systems: The Guard Dog, The Home Guard and the top of the line Terminator III. The assembly of the Guard Dog takes 1.5 hours per…
Amissadai Pinheiro
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The function $f(x,y)=\dfrac{x}{y}$

My concern is: Why the hessian of the function $f(x,y)=\dfrac{x}{y}$ is semidefinite positive by deducing from the quadratic form method, although, its determinant is negative ?? $$\nabla^2f(x,y)=\begin{pmatrix} 0 & \dfrac{1}{y^2} \\ \dfrac{1}{y^2} …
My question
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how to formulate the objective function

Formulate and solve algebraically, through optimal conditions first order, the problem of finding the point of the curve $x_{2} = x_{1}(3 − x_{1})$ that is closest to the point $(3 3)^{T}$. What is the guarantee that the point obtained is in fact…
Amissadai ferreira
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KKT conditions for a minimization problem

The author of the book was explaining the following question: Consider the following problem: $min -x_{1}$ subject to $ x_{2} -(1-x_{1})^3 \leq 0$ $-x_{2} \leq 0$Show that$x^{*}$ = $\begin{pmatrix} 1 \\ 0\end{pmatrix}$ it is a minimizer, but the…
Amissadai ferreira
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Linear Optimization under piecewise linear constraints

I have the following optimization problem. For given $r \in \mathbb{R}$, $y \in \mathbb{R}^{n \times K}$, $p \in [0,1]^K$ and $b>0$ and writing $c_{i} = \sum_{k=1}^{K}p_{k}y_{ik}$ : $$\max_{(x,\alpha) \in \mathbb{R}^{n} \times \mathbb{R}} \;…
Claudio Moneo
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find the global minimum

I have the following problem: Consider the quadratic function $f(x) = \frac{1}{2}x^{T}Ax + b^{T}x$ with $A \in \mathbb{R}^{n×n}$ symmetrical and $b\in \mathbb{R}^{n}$. Show that if $f$ is limited iinferiorly, then $A$ is semi-defined positive and…
Amissadai ferreira
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Non-convex cone and the Polar

I would like help with exercise: Be $S = \{ d\in\mathbb{R}^2\mid d \ge 0,d_1d_2=0 \}$. $(a)$ Show that $S$ is a non-convex cone; $(b)$ To determine $P(s) = \{p\in\mathbb{R}^2\mid p^{T}d\le 0, \forall d \in S\}$ $(c)$ Geometrically represent the…
Amissadai ferreira
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