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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maximization over a simplex

Suppose I am given two sets of real numbers $\{a_i\}_{i=1}^N$ and $\{w_i\}_{i=1}^N$ with $w_i>0$. I am trying to find the maximum of the expression $$\left\lvert \sum_i a_i \left(\frac{w_i s_i}{\sum_j w_j s_j}-s_i\right) \right\rvert$$ over a…
yml
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Can we minimize $3^m 2^{n/m}$, given $n$?

If we are given $n$, a positive real, can we find a the positive real $m$ that minimizes the function: $$3^m 2^{n/m}$$ I'd prefer to find the function that gives a value for $m$, but I'm also interested in asymptotic bounds for $m$.
Matt Groff
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optimal road layout problem - how to convert to maths and see the shapes it makes?

I have this puzzle going round my head about optimal road layouts, but I'm a programmer not a mathematician and I don't really know how to specify it as a maths problem. Once it's well-specified I can solve it (or more likely get help to solve it)…
jhabbott
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Maximizing and minimizing two functions simultaneously

Find $f$ which will maximize $y_1$ and minimize $y_2$: $y_1 = \dfrac{ I f}{ 1+a f} $ $y_2 = (bf^3 + cf)\dfrac{I}{y_1}$ Note : $I , a, b , c > 0 ; f > 0 $ Thanks.
marcella
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Objective function with inverse matrix

Let $\mathbf{v}\in \mathbb{R}^{p+1}$ a known vector and $\mathbf{A}\in\mathbb{R}^{p\times p}$, $\mathbf{B}\in \mathbb{R}^{n \times p}$ known matrices. In this setting, $\mathbf{A}$ is symmetric and invertible. My objective is to determine whether…
user326159
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Maximization problems - Using implicit function to avoid KKT

I am wondering if the following procedure to solve a maximization problem in - let's say - two variables with inequality and non-negative constraints works. More specifically, let's assume something like the following concrete example: $ \text{max }…
Kolmin
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Definition of a Closed Function

On page 9 of Convex Optimization Theory, a function $f: X \rightarrow [-\infty, \infty]$, $X\subset \mathbb{R}^n$ is defined to be closed if its epigraph $\mathrm{epf}(f)$ defined as: $$ \mathrm{epi}(f):= \{(x, w) |x\in X, w\in\mathbb{R}, f(x) \leq…
Door Handle
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Find the maximum

I would appreciate if somebody could help me with the following problem: Find the maximum of the function $$f(x,y,z) = x$$ on the curve defined by the equations $F(x,y,z) = G(x,y,z) =0$ with $$F(x,y,z):= x^2 +y^2 +z^2 -1 \qquad \text{and} \qquad…
user10938
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Optimum product disassembly / assembly path

I have a store Σ, which contains products (denoted by latin mayuscules) and components (denoted by greek minuscules). A product consists of a set of components (but not more than one of each distinct component). For instance: A = {β, γ}, B = {α}, C…
Hyperboreus
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Optimisation problem; Length of a cable

I am busy writing a program which determines the correct length and force on steel cables when hanging objects from the ceiling. One of the features is that it calculates the correct length, but I would like to add a feature which tells you which…
patrick
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The sum of two positive numbers is 1. The sum of their cubes is a maximum. What are the numbers?

I set this up and end up finding the minimum (the two numbers would both be $1/2$). To find a maximum value, I could reflect the functions and use $y^3-x^3$ but I still end up finding $1/2$ as the two numbers. It does make sense that two two…
user147485
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Resitor bank optmizer

I have a real world problem where the math is beyond me. I'm trying to set up an automated resistor bank much like a decade box. I have $18$ channel that I can switch in up to four channels in parallel at a time. Each channel is a fixed resistor…
CLampros
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Minimise the entropy of a probability vector using Lagrange multipliers

Problem statement: The entropy of a probability vector $ p = (p_1, ... , p_n)^T $ is defined as $ H(p)= - \sum\limits_{i=1}^{n} p_i \log{p_i} $, subject to $ \sum\limits_{i=1}^{n} p_i = 1 \mbox{ and } p_i \geq 0$, where $ 0\log{0} = 0 $ by…
Joshua Pepper
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Optimization on Stiefel Manifold

$$\text{Find}~~U, V$$ $$\text{to maximize}~~f(U,V)=\text{tr}(U^TAVN)$$ $$\text{subject}~~U^TU=I_p,V^TV=I_p$$ where $N=\text{diag}(\mu_1,\cdots,\mu_p)$ with $\mu_1>\mu_2>\cdots>\mu_p>0$. I am reading the book Optimization Algorithms on Matrix…
gaoxinge
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Minimize multi-variable function one variable at a time

I am wondering if I can minimize a multi-variable function one variable at a time. In other words, is it true that: $min_{x_1,x_2} f(x_1,x_2)=min_{x_1} min_{x_2} f(x_1,x_2)$
Tom Bennett
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