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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If $p,q,r$ are positive real numbers and $p+ q+ r = \frac{1}{2}$ , then find the range of $(1-p)(1-q)(1-r)$

If $p,q,r$ are positive real numbers and $p+q+r = \frac{1}{2}$, then find the range of $(1-p)(1-q)(1-r)$. Here, one can easily reach the upper limit, but what about the lower limit? Somebody please help me with this problem.
Catalan
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Minimise $\sum_{i=1}^n \frac{1}{a_i + x_i}$ subject to constraint using Lagrange multipliers

Minimise $\displaystyle \sum_{i=1}^n \frac{1}{a_i + x_i}$ subject to $\displaystyle \sum_{i=1}^n x_i =b$, $x_i\geq 0$ for $i=1,\cdots , n$, where $a_i >0$ for $i=1,\cdots , n$ and $b>0$. I know I need to use Lagrange multipliers to do this, but I'm…
user55225
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Optimisation problem choose x to minimize y

I have stumbled upon a sample maths question during my revision, and I have no idea how to solve it. Can anyone help or guide me along? Given a piece of rectangular paper of 11 cm by 8.5 cm. The lower left-hand corner is to be folded over to reach…
becozlah
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Trying to map my physics experiment onto an optimisation problem

I'm running an experiment consists in setting voltages on a set of 4 gates in an electronic device, and performing a measurement that will return a probability of success. My goal is to set the gates to a specific value that maximises the…
SuperNano
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What should I be researching to solve this problem?

This week's "riddler classic" on $538$ is quite the puzzler (as the riddler classic always is). However, I can usually deduce solutions by researching relevant material or just using what's already up in the 'ol noggin. The question…
Joseph Eck
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Minimize vector's L-infinity norm

Consider the equation $$\mathbf{v} = \mathbf{v}_0 + A\mathbf{x}$$ where $A$ is an $m\times n$ matrix ($m\gt n$) with entries equal to $-1$, $0$, or $1$ only. Additionally, $1$ and $-1$ only appear once in each row $A$ so the sum of a row is always…
user7926
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Golden Section Search With Noisy Measurements

I'm using a modified golden section search (brents) to find the maximum / minimum of a function. The function is a real time measurement from a laser that is measuring the height of a single peak on a piece of machined metal. The metal is machined…
bradgonesurfing
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Solving a system of $3 \arg \max$ equations

Considering these 3 equations: $$ \alpha_{1} = \arg \max\limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/\alpha}{1/\alpha + 1/2 \alpha_{2} + 1/2 \alpha_{3}} \big)^2 \Big \} \\ \alpha_{2} = \arg \max \limits_{\alpha \geq 0} \quad…
juju
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Under what circumstances is it acceptable to assume all variables will be equal in optimization problems?

I'm sorry for the general and kind of inaccurate title, but I'm not sure how to phrase this question in a super short way. In calculus class a few years ago, this question was given: Choose three positive numbers $x, y,$ and $z$, so that $x + y + z…
Polygon
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Joint/Simultaneous optimization

$\DeclareMathOperator*{\argmin}{arg\,min}$ Suppose I have to jointly minimize two functions. The solution to the joint minimization does not necessarily minimize each function individually but sort of best one can get for both cases together. e.g.,…
AbstractDissonance
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The maximum value?

$x^5y + y^5z + z^5x$ Let $x$, $y$, and $z$ be non-negative reals such that $x+y+z=1$ The maximum value of the above expression can be represented as $a^b/c^d$, where a and c are not perfect powers, and $a, b, c, d$ are positive integers. Find the…
Ahmed Amir
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Why are closed feasible sets preferred, from an optimization point of view?

The title states my question: what aspect of closed makes it attractive for optimization?
Blackjaguar
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Does the opposite of the brachistochrone exist?

Suppose a particle only influenced by gravity is going from point a to point b on an x,y plane. Is there an equation in which the particle takes the longest amount of time, yet still makes it to point b? I know the equation for brachistochrone is…
Marcus
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Finding saddle point of a quadratic form

I am trying to find the saddle point of a quadratic form: $$f(\mathbf{x})=\mathbf{x}^\mathrm{T} \mathbf{A}\mathbf{x}+\mathbf{x}^T\mathbf{b}+\mathbf{c}$$ using a minimization/maximization-like algorithm. Is there such solution method available…
pluton
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Computing the volume of a set efficiently

Given a set of vectors $\mathbf v_i$ for $i=1,\dots,k$, $\mathbf v_i \in \{0,1\}^n$, is that possible to efficiently find the volume of the set, $$\left\{\mathbf x \in [0,1]^n:\mathbf x \le \sum_{i=1}^k \alpha_i\mathbf v_i\ \text{for some…
Helium
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