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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Minimizing weighted sum of distances

given a set of coordinates and the following function: cost = $\sum \sqrt{(x_i−X)^2+(y_i−Y)^2}w_i$ I would like to find the point (X, Y) for which this function is minimal. A simple example shows that the weighted average of the coordinates does not…
Dieter
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$\max_x \max_y f(x,y) = \max_y \max_x f(x,y)?$

Just come across a question regarding sequential maximization and simultaneous maximization, and I do not recall whether there are any established conditions for the equivalence. Anyone has some idea? $$\max_x \max_y f(x,y) =\max_y \max_x f(x,y)?$$
Confusian
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Minimize $\| Ax-b \|$

$A$ is a $n \times m$ matrix with known real elements and $b$ is a known real $n$-dimensional vector. I would like to find all $x$ such that $\| Ax-b \|$ is a minimum. Is there a theorem that deals with it? Update: What changes if we add the…
tchronis
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Maximising with multiple constraints

I have $$Z=f(x_1 ,x_2 ,x_3 ,... ,x_n)$$ function and $$\left[\begin{array}{r}c_1=g_1(x_1 ,x_2 ,x_3 ,... ,x_n) \\c_2=g_2(x_1 ,x_2 ,x_3 ,... ,x_n)\\c_3=g_3(x_1 ,x_2 ,x_3 ,... ,x_n) \\...\\c_m=g_m(x_1 ,x_2 ,x_3 ,...…
Huseyin
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Basic Question about Newton's Method for Optimization

This is a very basic question about Newton's method for optimization, but I cannot seem to find the answer in any of my searches. If we are using Newton's method (or gradient descent), how do we find a maximum instead of a minimum? Do we just…
user1893354
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Compute the minimum distance between the centre to the curve $xy=4$.

I wish to solve the following problem: Compute the minimum distance between the center to the curve $xy=4$. But I don't know where to start from?
Niousha
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Minimum value problem

Find the minimum value of $(x+y)(y+z)$ where $x,y,z$ are positive real numbers satisfying the condition $$xyz(x+y+z)=1$$ Hint?
user85798
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Equivalence of Maximizing Products, Sums, and Sum of Logs

Throughout this question assume $f_i \ge 0 \forall i $. I know that for any (single) function the following is true $$f(x^\star) \ge f(x) \text{ }\forall x\in X$$ iff $$\log(f(x^\star)) \ge \log(f(x)) \text{ } x\in X$$ Now suppose fundamentally we…
Squirtle
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Minimization of a function including quadratic and exponential terms

Apologies for the simple question. Is it possible to analytically minimize a function such as $$f(x) = a^x + c x (x-1) \qquad x > 0, \quad 0 < a,c < 1$$ Nothing else can be said about the constants $a$ or $c$.
user1910
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Why is lasso not strictly convex

I know a nonmonotonic convex function which attains its minimum value at a unique point only is strictly convex. I didn't get how lasso is not strictly convex. For eg if I consider two dimensional case. $||x||_1$ attains its minimum value at (0,0)…
user34790
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Rate of Convergence for (Gauss)-Newton method

I am trying to solve a non-linear least squares problem with newton's and gauss-newton method. $\min \left\Vert \pmb F( \pmb x) \right\Vert_2^2$ with $F: \mathbb{R}^m \rightarrow \mathbb{R}^n$. Furthermore, I want to analytically calculate the local…
bonanza
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Optimizing a meal based on the nutrients in the foods in it

ladies and gentlemen. I am not a mathematician, so go easy on me. I'm a programmer. I have a database that I got from the Internet (USDA National Nutrient Database for Standard Reference), detailing the amount of each nutrient in each of a few…
Tiiba
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Existence of global minimum of $f(x) = \sum_{i=1}^n e^{x_i}$

Let $K$ be a subset of $\mathbb R^n$ satistifying the following properties: $K$ is a linear subspace of $\mathbb R^n$ If $x \in K$ either $x= 0$ or there exists an $i \in \{ 1, \ldots n\}$ such that $x_i >0$. Does $f(x) = \sum_{i=1}^n e^{x_i}$…
Ben Martin
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Extrema homework — maximizing the viewing angle of a picture on a wall

I have hit a problem in my homework and don't know how to solve it. Here it is: "A picture with height of 1.4 meters hangs on the wall, so that the bottom edge of the picture is 1.8 meters from the viewers eye. How far does the viewer have to stand…
user63021
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Scaling of objective function in optimization problems.

In a lot of optimization algorithms I have seen that they use a scaling factor to scale the objective function. I dont understand what is the reason behind this. Why do we need scaling of objective function in optimization algorithms? Does it work…
Student
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