Mathematics Stack Exchange
Mathematics Stack Exchange

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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Can all Lagrange multipliers be non zero?

I want to know that can all of the lagrange multipliers can be non zero at the same time. I have a problem with two variables and three constraints. I solved the problem for all possible combinations of the multipliers i.e all multipliers are zero,…
shahid
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$f(x_1,x_2)=x_1x_2$ in $\mathbb R^2_{++}$, positively definite?

The square form is $H:=x^T\nabla^2 f(x) x= 2 a b$ where $x=[a,b]$. Now $f(x_1,x_2)=x_1x_2$ in $\mathbb R^2_{++}$ (problem b). I am perplexed: I think my teacher means that this not positively semidefinite because $H>0$ -condition is not satisfied…
hhh
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Loss optimization of $max$ function

I have to find the function $f(x)$ that minimizes the expression $$ L(y, f(x)) = \left\{\begin{array}{ll} \max(0, 1 - yf(x)), & yf(x) \geq 0 \\ 1 - kyf(x), & otherwise\end{array}\right.$$ where $y \in \mathbb{R}$ and $k \geq 1$. I know…
Taufi
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Optimizing over a dinamic multi-dimensional space

I would like to optimize a function $f(x): \displaystyle\bigcup_{n=1}^\infty \mathbb{R}^n \rightarrow \mathbb{R} $. It seems that classical optimization routines does not work here since the space does not have a fixed dimension and it is…
Francisco
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Comparing Equations to find optimum point

If I have 2 or more eqiations of degree 5, each representing a parametrr that I'd want to optimie. Say, one equation shows trend of production and the other one trend of rejections and both are function of $x$. Now I need a value of $x$ at which my…
Bilal Anjum
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How to show that the function is concave when second derivative is inconclusive ?

How can I show that the following function $f(\lambda)$ is concave for $\lambda \geq hz_1 + \sigma^2$ ? $f(\lambda) = 1- \exp(\frac{hz_1 + \sigma^2 - \lambda}{z_1b}) + \exp(\frac{hz_0+\sigma^2-\lambda}{z_0b})$ where $z_1 > z_0$, $b>0$ and $\sigma^2…
King008
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Evaluate error on a oscillatory curve?

since I am trying to find the fitting parameters for a model, sometimes I get this: enter image description here basically I have to have that the green curve (my model) at certain point should stop to oscillate, like the target (the blue one). The…
Italo Persechino
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Maximizing Profit given just Cost Function

To clarify this is for an economics course. $$ x \in \mathbb R^n $$ For a firm with two outputs with cost function where $$ C(q)=q_1^2+q_2^2+10 $$ given output levels $q=(q_1,q_2) \ge 0$. If output prices given and fixed, find profit maximizing…
Shaner
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Minimize linear function with multiple variables

So this seems like a very simple problem but I am having trouble figuring out the best approach. I may also be formulating the problem incorrectly with additional constraints I'm forgetting to explicitly state. I need to do the following (2…
Kev
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Why do all points in $\Omega$ satisfy the FONC for this set-constrained problem?

First-Order Necessary Condition (FONC) Let $\Omega$ be a subset of $\mathbb{R}^n$ and $f\in\cal{C}^1$ a real-valued function on $\Omega$. If $\mathbf{x}^*$ is a local minimizer of $f$ over $\Omega$, then for any feasible direction $\mathbf{d}$ at…
Danny_Kim
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How to build the Lagrange function when variable contain same part?

Recently, I encoutered the following objective function $$ \text{minimize}\quad \mathbf E(\mathbf X)=\mathbf X^{\text T}\mathbf K \mathbf X\\ \text{s.t.} \quad\mathbf A\mathbf X=\mathbf P $$ where $\mathbf X=[x_0,\cdots,x_{\hat n+p}]^{\text…
user123
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Time constrained optimization methods

I am interested in optimizing a function $f(x)$ but evaluating the function for some $x$ is costly. Each evaluation of the function takes a time $t$ and I have a total time $T$ in which to attempt to optimize this. Has there been research into how…
Hugh
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An absolute minimum in the plane?

The origin is a critical point of the function $f(x,y)=x^2+x\sin y+\log(1+y^2)$. Moreover, it is a relative minimum because of hessian matrix at that point is $\begin{pmatrix} 2 & 1\\ 1 &2\end{pmatrix}$. The question is: is the origin an absolute…
Ana María CG
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Proving generalized Weierstrass theorem

Let $f:X\to \mathbb{R}$ be a lower (upper) semi-continuous function and $X$ is compact. Prove or disprove: $$f \text{ is bounded from the top and bottom,}$$ $$\text{There exists }\min_{x\in X}f(x) \text{ and } \max_{x\in X}f(x),$$ $$\text{ For each…
user368075
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Most surface area enclosed by fences

There two roads that interesect at an angle of 68 degrees, how should he put two fences(line segments) length L such that maximum area is between fences and road. I'm guessing at a 90 degree angle between the fences?
user354021
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