Mathematics Stack Exchange
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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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How to find the maximizer in the Legendre transform of cumulant generating function?

Cramer's theorem in the large deviations theory gives the rate function $\sup_{\boldsymbol{\lambda}} \left<\boldsymbol{\lambda},\,\mathbf{b}\right>-\log\int_{\mathbb{R}^n} \exp(\left<\boldsymbol{\lambda},\,\mathbf{x}\right>)\,P(dx)$, which is the…
Erik Apostol
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Local minimum of the function

We have a function $$\sum_{i=1}^{n}x_{i}^{2}=min,\quad\mbox{ subject to }\sum_{i=1}^{n}x_{i}=c,$$ which should have the minimizer $$\frac{c^{2}}{n}.$$ However, from the…
tomb
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Minimizing $L_\infty$ norm using gradient descent?

Curve fitting problems are solved by minimizing a cost/error function with respect to the model's parameters. Gradient descent and Newton's method are among many algorithms commonly used to minimize this function. The $L_\infty$ norm can also be…
rookie
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Optimization for fat line intersecting most points

Let's say I have a bunch of $(X,Y)$ points in 2D space. I want to find the line $y=mx+b$ which intersects the most points. We can add some kind of buffer (a delta) so if the line $y=mx+b$ is within delta of the point, then it also intersects the…
Michael
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Optimization for non linear function

I have an injective function $f:N\mapsto \mathbb{R}^k$ with $N\subset \mathbb{R}^{k-d}$ open. Let $v_n\in \mathbb{R}^k$ is a sequence of random variable. I'd like to find $\hat x\in N$ such that \begin{equation*} \|f(\hat x)-v_n\| \end{equation*} as…
Jlamprong
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Optimization/Jensens Inequality

A community has a fixed stock $X$ of oil that it has to consume over an infinite horizon. The utility function to be maximized is $$U=∑_t \delta^t \ln (C_t)$$ where $C_t$ represents consumption of the resource at time $t$. Also $\delta_t \in [0,1]$…
kangkan
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Is my use of Lagrange multipliers correct?

Given $n$ positive values. $x_i \ge 1 (1\le i \le n)$. Their sum is $k$. $$ \sum_{i=1}^{n}x_i = k $$ Define the following value: $$ \sum_{i=1}^{n}x_i(x_i-1) $$ Now use Lagrange multipliers to find when this value is minimum. Define $J =…
Jingguo Yao
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Minimizing the distance between two sums of ratios

I have an optimization problem which I think might be fairly difficult, so I want to ask if it's practically possible. To start out, I have a linear programming problem, for which I obtain the optimal objective value. The problem has multiple…
jarlemag
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Find and Evaluate Critical Points

I need to find al the critical points of the following function $f(x,y)=y^2-x^2y-3y+x^4-x^3$ Determine if they are local minima, local maxima, or saddle points, by looking at the Hessian matrices at the critical points.…
user2553807
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Classical Newton method for minimization

For a quadratic convex function Classical Newton method for unconstrained optimization reaches the minimum point in one iteration. It this true? If so, what is the proof ?
user107723
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Preconditioning of optimization problems

This question suggests that you can precondition an optimization problem by a simple multiplicative scaling of the variables in the objective function. However, when I look up literature on preconditioning I see it is typically defined on matrices…
Bitwise
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Convex hull of a 0/1 set in $ \mathbb{R}^d$

Reading in my textbook, I found the following example: $$ $$Let S $ \subseteq $ {0,1}$^d$ be an arbitrary 0/1 set in $ \mathbb{R}^d$ and the Polyhedron Q = conv(S). It can be shown easily that the set of the vertices of Q is equal to S. I don't find…
user120016
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What optimization method should I use here?

Suppose that we have the following (unconstrained) optimization problem: $$ \min_{\mathbf{w},b} \frac{1}{2}\Arrowvert\mathbf{w}\Arrowvert^2 + C\sum_{i=1}^{l}g_i(\mathbf{w},b), $$ where $\mathbf{w}\in\mathbb{R}^n$, $b\in\mathbb{R}$. Moreover, for…
nullgeppetto
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Method for the minimum of $x\cdot\sin(1/x)$?

I have the function $f(x)=x\cdot \sin\Bigl(\dfrac{1}{x}\Bigr)$ $(\mathbb R^*\rightarrow \mathbb R)$. Is there a method to find the global minimum of $f$? Thanks
user113865
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Counterfeit coin problem.

Problem $N$ coins, $N-1$ equal coins and one heavier counterfiet coin. With a balance beam given we want to find the counterfeit coin. We have two different goals: minimise the expected number of weighings minimise the maximal number of required…
Nedellyzer
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