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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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How to figure out the contribution of each factor?

Suppose we have value $S$ and then we multiply it by $xyz$. So, $$T = Sxyz$$ How can we figure out how much each of $x,y,z$ contributes to $T$? One way is to do: $x$ contribution: $Sx - S$. $y$ contribution: $Sxy - Sx$ $z$ contribution: $Sxyz -…
steviekm3
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Conditions for unique, infinite, and no solutions of optimization problem

I have attached the question I am referring to. I assume for the uniqueness one, we can say that the condition is that it has to be a convex function with a global minima. But that does not seem right. And I'm not sure at all about the infinite and…
Aurangzeb Rathore
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There exist a point which is local minimizer in all directions, but not local minimizer

I'm looking for such counter-exemple. Several times I get me thinking in situations that if I was aware of such exemple I would get rid of future problems. Is there such counter-exemple? Nomenclature: $x$ is a local minimizer in direction $d$ of…
R. W. Prado
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How to show that $g(y):=\frac{1}{2}\sum_{i=1}^n\left(\left(\sqrt{\lambda_i}y_i -c_i\right)^2-c_i^2\right)$ is strictly convex?

This question is a continuation from the question: Show there exists a global minimum In the accepted answer, the solution continued to: $g(y):=\frac{1}{2}\sum_{i=1}^n\left(\left(\sqrt{\lambda_i}y_i -c_i\right)^2-c_i^2\right) $ I understand how to…
Rico
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Is maximizing $f = {X^T}{A^{ - 1}}X$ same as maximizing $f = {X^T}{\left( {A + {\sigma ^2}I} \right)^{ - 1}}X$ subject to the constraint $X^{T}X=1$

Where $X$ is an $N{\times}1$ vector, $A^{-1}$ is a $N{\times}N$ symmetric positive semi-definite matrix, and the elements of A=x[n] @x[-n],where A is the auto-correlation matrix, where @ means the convolution operator. $\sigma^2$ is a constant and…
Jordan
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Find two circles with smallest area to cover all the given points on Cartesian plane.

Problem: Find two circles with smallest area to cover all the given finite points on Cartesian plane. Each circle should cover at least two points. For the simpler problem, when we need to find just one circle to cover all the given points, I know…
Lee
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Formulating an Optimization Problem

Can you please help in formulating the following optimization problem: Consider the problem of finding an $x\in\mathbb{R}^n$ that satisfies at least $k$ (with $1\le k\le n$) of the following $m$ constraints: $$a_i^Tx\le b_i,\quad\text{for }…
Salwa
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$ \min_{x} [g(x) + h(x)] = \min_x g(x) + \min_{x} h(x)$ proof?

So given $$\min_ \left.x\in[a,b \right] \left[g(x) + h(x) \right]= \min_{x\in[a,b]} g(x) + \min_{x\in[a,b]}h(x),$$ I am asked to prove or provide a case where this doesn't hold. I should also mention that x $\in$ [a,b] and [a,b] -> $\mathbb{R}$. I…
Gordqir
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Optimal monthly contribution towards debt vs. savings

Starting with: \$0 in savings and \$50000 in debt the savings earns 2.5% interest and the debt loses 5% interest yearly you gain \$2000 of income each month to distribute among either savings or debt (fully committed to either so there is \$0 in…
AAC
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Improving optimization speed by obtaining upper bound of objective function?

For example, when I am trying to minimize the function $f(x) = x^2 - 6x +12$ where $x\geq$ are non negative real number. If by some way, I was able to obtaining the upper bound (only) or lower bound (only) of this problem but without knowing the…
Tuong Nguyen Minh
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Question regarding optimization using Lagrange multipliers

Let $K=\{(x,y):x^2+y^2=1\}, A=\{(x,y,):2x+3y=10)\}$ Find $x \in K, y\in A$ sucht that $d(x,y)=d(A,K)$. The lagrangian function is…
user700332
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Optimization problem with range of variables

I have a simple optimization problem. The objective function consists of $4$ variables, say $a,b,c$ and $d$. So the objective function $y=f(a,b,c,d)$ is a linear function of $a,b,c,d$. The constraints for these variables are only their range,…
Shishir
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Iterative algorithm to find the optimal solution of multi-variable optimization problem

I have a optimization problem presented as following: \begin{equation} \mathop {\max }\limits_{({x_1} \to {x_5})} f({x_1},{x_2},{x_3},{x_4},{x_5}) \end{equation} where $f(x_1, x_1, x_3, x_4, x_5)$ is a multi-variable function. It is impossible to…
AT170
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What is bounded by $O(T)$ mean?

I have come across something like $f(T)$ be bounded by $O(\sqrt{T})$ many times in optimization context. usually, $f(T)$ don't have an explicit form. If $f(T)$ do have an explicit form, say $f_1(T)=\frac{1}{2}\sqrt T$ and $f_2(T)=\frac{3}{2}\sqrt…
dawen
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Integer optimization problem

Suppose we are given $Av - x \ge 0$, for a given $n \times n$ matrix A and an $n\times 1$ vector $x$. Find an integer valued vector $v$ of size $n \times 1$ such that $\mathbf{1} \cdot v$ is minimized (in other words the sum of its elements is…
darksky
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