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").

22512 questions
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How to prove the sum of squares is minimum?

Given $n$ nonnegative values. Their sum is $k$. $$ x_1 + x_2 + \cdots + x_n = k $$ The sum of their squares is defined as: $$ x_1^2 + x_2^2 + \cdots + x_n^2 $$ I think that the sum of squares is minimum when $x_1 = x_2 = \cdots = x_n$. But I can't…
Jingguo Yao
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Math Wizardry - Formula for selecting the best spell

Imagine we have a wizard that knows a few spells. Each spell has 3 attributes: Damage, cooldown time, and a cast time. Cooldown time: the amount of time (t) it takes before being able to cast that spell again. A spell goes on "cooldown" the moment…
aaronfarr
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Global maximum of a sum of Gaussians

Suppose I have the weighted sum of $n$ Gaussian functions with varying means and standard deviations, $$f(x) = \sum_{i=1}^n a_i\exp\left(-\frac{(x-\mu_i)^2}{2\sigma_i^2}\right).$$ All the weights $a_i$ are positive. I want to find the position of…
user856
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Conceptually, why does a positive definite Hessian at a specific point able to tell you if that point is a maximum or minimum?

This is not about calculating anything. But can anyone tell me why this is the case? So, from wikipedia: If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local…
rjm726
  • 151
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Is the layout of Burning Man's city "grid" optimal?

My intuition is that the layout of Burning Man's city "grid" optimizes for the smallest sum of all distances between any two points on the map. Am I correct? Is the proof obvious? Or is there another way to lay out a city to minimize the distance…
Robert Atkins
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What are active constraints?

I am asked why the constraint $x_1\leq 2$ would be active when maximizing $$ 8(x-1)^2 +2(y-1)^2 $$ subject to $$12x+12y=126$$ But I am not sure what it means for a constraint to be active. We are using Lagrangians to do this.
Demetri Pananos
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Making sense of the big world of gradient methods

There are many extensions of gradient descent: stochastic-, Nesterov accelerated-, proximal-, conjugate-, dual-, mirrored-, splitted-, coordinate- gradient descend and more. It also appears that many of these can be combined (there is a paper titled…
Jules
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9
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Minimize exponential function over an interval

I want to minimize the function $$f(x) = e^{ax^2} + e^{b(1-x)}$$ with respect to $x$ (where $a$ and $b$ are constants), subject to the constraint that $0 \leq x \leq 1$. I know that $x = 0$ and $x = 1$ are both critical points, but I am interested…
jamaicanworm
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Forming two three-digits and one two-digits to maximize product

So I am helping a grade 5 student to prepare for an olympiad and found this question from the olympiad's past paper : Using $1, 2, 3, 4, 5, 6, 7$ and $8$ (without repetition), form two three-digit numbers and one two-digit number such that the…
Ricky The Ising
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Splitting equilateral triangle with shortest curve

A colleague of mine just told me this amusing optimization problem: Given an equilateral triangle, cut it into two pieces of the same size with a curve of minimal length. I only want some insight or hint that might help me solve this. I am thinking…
Tal-Botvinnik
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8
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1 answer

Dynamic programming - A type of balanced 0-1 matrix

I was reading the Wikipedia article of Dynamic programming, however, I'm having a hard time understanding the explanation given in the example for a type of balanced 0-1 matrix. The problem is stated as: Consider the problem of assigning values,…
r_31415
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Dual problem of quadratic program general theory

At the moment I'm studying constrained optimization. Recently we had a lecture concerning dual problems. I have a few questions regarding this. The primal problem is given as \begin{align} \min_x \quad & \frac{1}{2} x^THx+g^Tx \\ \text{s.t.}…
ANYN11
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7
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To minimize $x^TAx$ where $A$ is not necessarily positive semi-definite.

Let $A\in \mathbb{n\times n}$ be a symmetric matrix. Let $x\in \mathbb{R}^{n\times 1}$ be an unknown vector. The problem is $$\min \limits_x x^TAx.$$ Since $A$ is an input, I am not sure 1 it is positive semidefinite (the objective is convex); 2…
user18481
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1 answer

Questions about constraints and KKT conditions

KKT conditions from Wikipedia: We consider the following nonlinear optimization problem: $$ \text{Minimize }\; f(x) $$ $$ \text{subject to: }\ g_i(x) \le 0 , h_j(x) = 0 $$ The number of inequality and equality constraints…
Tim
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In constrained optimization problems, when is 'naive' substitution possible?

To motivate the question, consider the following constrained optimization problem: $$ (P1)\quad \underset{(x,y)}{\min} f(x,y)=x^2 +y^2 \ s.t.\ (x-1)^3 = y^2$$ By replacing the constraint $y^2 = (x-1)^3$ in $f(x,y)=x^2 +y^2$, one gets the following…
shamisen
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