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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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Relationship of the Solution Set of a Given Optimization Problem to Another Given Set

How is the solution set of the optimization problem $$\min \frac{1}{2} x^T A x - x^T b \text{ s.t. } x \in K,$$ where $A$ is symmetric positive semi-definite and $K$ not convex related to the set $$\{x\ |\ \forall i : x_i = \operatorname{arg min}_{x…
user1225999
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Minimum number of operations

We have a sequence $A$ of size $n$, where each element $A_i \in \{-1,0,1\}$. In each operation we can increase $A_{i+1}$ by $A_i$. The goal is to make the sequence non-decreasing i.e. $A_1 \leq A_2 \leq \dots \leq A_n $ with the minimum number of…
Gary
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What is the Minimal cost?

A power house, $P$, is on one bank of a straight river $W$ meters (m) wide, and a factory, $F$, is on the opposite bank $L$ meters downstream from $P$. The cable has to be taken across the river, under water at a cost of $\$CW/\mathrm{m}$. On land…
marchetto
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Optimization problem formulation

The problem asks me to formulate the following optimization problem, find the first order conditions, solve them and verify second order conditions. So the problem is the following. I have to maximize the volume of a cardboard box, given that the…
user191919
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maximization over a max operator

My question is as follows \begin{equation} \max_W\max_{D_k}D_kW \end{equation} where \begin{equation} D_k = ((x_{i}^{1}-x_{j}^{1})^{2},(x_{i}^{2}-x_{j}^{2})^{2},...(x_{i}^{n}-x_{j}^{n})^{2}) \end{equation} $D_k$ is a $1 \times n$ vector and $W$ is…
arescat
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optimization over equations with max,min

Notation: $\max( x_1, \cdots, x_n )$ denotes the maximal number among $x_1, \cdots, x_n$. $\min( x_1, \cdots, x_n )$ denotes the minimal number among $x_1, \cdots, x_n$. Assumption: $x_i, a_i, b_k$ are all in $[0,1]$ minimize $\max( \{…
user124986
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If $0\le x\le 1$ and $0\le y\le 1$, find $\max\{(x^2y-y^2x)\}$

If $0\le x\le 1$ and $0\le y\le 1$, find $\max\{(x^2y-y^2x)\}$ My work: Though I could not approach the problem, I tried to find out a few facts. So,I defined the above expression as $f(x,y)=x^2y-y^2x$. Now,I see that whenever $(x,y)=(k,k)$ the…
Hawk
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Maximizing a given function?

I have a function as below, $f(\alpha) = \frac{{1 - \alpha }}{2}\ln \left( {1 + \frac{{AB}}{{B + \frac{{1 - \alpha }}{{C\alpha }}}}} \right)$, where $A$, $B$, $C$ are constant, and $0 < \alpha < 1$ is a variable. Can someone give me a hint how to…
BinhDDT
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How to do optimization on expression which includes Reciprocal

The conditions are: $w$ is a known value, and $x_{11} >0, x_{12}>0, ..., x_{nn}>0;$ \begin{equation} x_{11} \leq w \end{equation} \begin{equation}x_{12} + x_{22} \leq w\end{equation} \begin{equation}x_{13} + x_{23} + x_{33} \leq…
user3007260
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Reasons for the worst-case scenario in robust optimization

When we solve an optimization problem, containing in his objective function an uncertain parameter (i.e. random variable), using robust optimization techniques such as the max-min approach, we first solve the problem for the worst case of the…
omar
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Definition of stability in the case of Levenberg-Marquardt optimization method

I've come across this guide: Fortunately, it inherits the speed advantage of the Gauss–Newton algorithm and the stability of the steepest descent method. What's a stability in this case? Does it mean that for small variation of input the output…
m0nhawk
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Propose suitable algorithm for min-max optimization problem

Consider: \begin{equation}\min_{x, y} \max_{\omega} | f(x, y, \omega) |\end{equation} where $(x , y)\in \mathbb{R}\times \mathbb{R} $ and $\omega \in (0, \infty)$. $f$ is the result of dividing two polynomials. In exact details we have…
Zia
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Help solving an optimization problem involving inverse square roots

Does anyone know if the following optimization problem has an elegant solution? Let $A=\{a_1, a_2, \ldots, a_n\}$ be positive real numbers. Let $B=\{b_1, b_2, \ldots, b_n\}$ be unknown positive real values that sum to a fixed total $L$. Given $L$…
Tom Dietterich
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Optimization problem of quadratic function - Compressive sensing

I got the optimization problem in Compressive sensing in form $f =arg min \ \frac{\mu}{2} ||\Phi.f-y||^2 + \frac{\lambda}{2} ||f-v-w||^2$ where $\Phi$ is orthogonal Gaussian sensing matrix size $M\times N, M << N$ and all vector $f, v, w$ with…
Atena Nguyen
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hessian matrix and conditions for semidefinite, definite

In the lecture notes this was given as a theorem. Let A be a $n \times n$ matrix and let A$_k$ be the submatrix of A obtained by taking the upper left hand corner $k \times k$ matrix of A.Furthermore let $\det(A_k)$,be the kth principal minor of…
clarkson
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