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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Maximize linear combination

Given a set of positive values $v_1, v_2, ..., v_n$ and a set positive of factors $\alpha_1, \alpha_2, ..., \alpha_n$, both ordered increasingly, show the maximum linear combination you can get is $$\sum_{i=1}^{n}\alpha_i \cdot v_i$$
cronos2
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Maximizing profit (dynamic programming)

I'm looking at a dynamic programming question and can't figure out how to solve it. The question is listed at the following website (question number 19, towards the bottom). http://web.mit.edu/15.053/www/AMP-Chapter-11.pdf The question asks to find…
AAKK
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Given $R \in \mathbb R$, choose $a,b,c$ from discrete set so that $a^{-1} + b^{-1} + c^{-1} \approx R^{-1}$

I am working with the following equation (parallel resistors): $\frac{1}{R_g} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$ The values of $R_1, R_2$ and $ R_3$ are discrete - lets say 256 steps in increments of 3: $$R_1,R_2,R_3 \in \{3n : 1 \leq…
Erik5000
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Question about separability of convex envelopes

Given a function $f(\boldsymbol{x})$ defined on the hypercube $\boldsymbol{x} \in [0,1]^n$. Suppose $f(\boldsymbol{x})$ can be expressed as $f(\boldsymbol{x})=c(\boldsymbol{x})+g(\boldsymbol{x})$, where $c(\boldsymbol{x})$ is a convex function and…
Golabi
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Finding Maximum and Minimum for f(x,y)

The problem I am working on is: Find the maximum and minimum values of the function: $f(x,y) = -3x^2 - 14xy - 3y^2 -8$ on the disk: $x^2 + y^2 \leq 4$ The $-14xy$ term is severely throwing me for a loop. It's complicating finding the critical points…
Kommander Kitten
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Optimization problem for a rectangle with the greatest possible area.

A rectangle is inscribed with its base on the $x$-axis and its upper corners on the parabola $y=7−x^2$. What are the dimensions of such a rectangle with the greatest possible area? Would this be a basic optimization problem, with the constraint…
Evan
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How should I treat a linear relaxation to a rucksack optimization problem?

I am currently studying for an exam in optimization, and in one of the questions the following was mentioned: "The linear relaxation of the problem $(P)$: $z^*=\max 4x_1+11x_2+7x_3+13x_4+15x_5+17x_6$ when $x_1+3x_2+2x_3+4x_4+5x_5+6x_6 \leq 9$ and…
Scounged
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Would I apply max/min to find out how many of each bird to get 200 birds for $200?

You are off to a bird market with \$$200$ to purchase $200$ birds. The following are prices of birds in the market $40$ Pigeons — \$$2$ $2$ parrot — \$$2$ $2$ falcon — \$$10$ If you have to spend the entire \$$200$ and you are not supposed to buy…
imparante
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converting from max to min in an optimization function?

I have the following maximization objective function related to a svm \begin{align} \max_{\gamma,\omega,b} & \ \frac{\hat{\gamma}}{\|\omega\|}\\ \mbox{s.t. } & \ \ \ y^{(i)}(\omega^{T}x^{(i)} + b) \geq \hat{\gamma}, \ \ i = 1, \ldots…
Lila
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Update step in gradient descent

Suppose I have a function $f(x)$ for which I want to find minimums. I understand that differentiation with respect to $x$ will give direction $+/-$ in $x$ axis to follow in order to minimize. Choosing a fixed step is understandable, but why do we…
joseph
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Difference between mean-variance and worst-case optimization for normal distribution

I have two optimization problems. 1-) Mean-variance optimization $J_{MV} = J_M - \gamma J_V$, where $J_M$ is mean, $J_V$ is the variance term and $\gamma$ is the weight on variance term. 2-) The worst-case optimization: $J_{WC} = max_u min_\theta …
Mohsin
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Global Max and Mins

Given $f(x,y) = x^2+y^2-14x-20y$ and restrictions $x≥0,\; 0≤y≤42 \;\text{and}\; y≥x$, I need to find the max and min. By finding partial derivatives and setting them to $0$, I get the min to be -149. However, I'm having trouble finding the max.
Lauralolo
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Another naive question: Objective in (maximization) optimization increases and decreases during iterations!

I have following optimization problem: $\max_{u,z} z$ $s.t \quad z - J_i(u,\theta_i) \leq 0 \quad\forall i$ $z$ is a scalar, $J$ is a nonconvex,non linear function of $u$ and $\theta_i$ is just uncertainty ($\theta_i$ shows the $i^{th}$ scenario).…
Mohsin
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Does this optimization, $\min n_1+n_2 ~\text{s.t.}~ \frac{9}{n_1}+\frac{2}{n_2} \le \epsilon$, make sense?

Suppose $n_1$ and $n_2$ are positive integer numbers. How can I approach this problem: $$\min n_1+n_2 \\~\text{s.t.}~\frac{9}{n_1}+\frac{2}{n_2} \le \epsilon$$
Elrond Gimli
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Steepest-descent optimization method

I was wondering if the steepest descent method can find a global min/max or only local min/max? Thanks
user161260
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