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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").

22512 questions
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Simple optimization problem.

In my Calculus I assigment, I'm stuck on the following : Find $M_1=(x_1,y_1)$ on $y=5x+6$ and $M_2=(x_2,y_2)$ on $y=-(x-3)^2+4$ such that the square of the distance between $M_1$ and $M_2$ is minimal. I'm fine with that, but they add the following :…
user111288
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Optimization problem where convex and non-convex problems are " mixed "

I have a naive question for an optimization problem. Let $$F(x_1,x_2,x_3)=(a-x_1 \frac{(1-x_2)\sin x_3}{1-x_2 \cos x_3} )^2 + (b-x_1 (1-x_2^2))^2$$ where $a$ and $b$ are known values, I have an optimization problem that consists in minimizing …
user94677
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Feasible direction for a point

I am slightly confused with what I am asked to do when trying to prove that a boundary point satisfies the first order necessary conditions for it to be a minimum, ie that the dot product of the gradient of the function at the point and any feasible…
elaRosca
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Minimization problem using PMP

Problem: I have to give the minima, subject to $x'(t) = u(t)$, $x(0) = 0$ and $x(1) = 1$ of the following function: $\int_0^1 u(t) dt$ I have to find this minima using the Pontryagin's Maximum Principle. What we have so far: $H = pu- \lambda_0…
Nedellyzer
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Maximize expected return

making a practice exam I had to make the following problem which I couldn't solve unfortunately... Problem In the springtime, a student has $N$ days to find a summer job for one month. Each day, the student is offered a job with total salary drawn…
kees den otter
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Minimum of $\prod_{1\le i\le n} (1+a_i)$ when $a_1a_2\cdots a_k = M.$

I asked a question here, and also got its generalization:(see tc1729's answer) $$\prod_{1\le i\le n} (1+a_i)\ge 2^{n}\sqrt{a_1a_2\cdots a_n}=2^n\cdot\sqrt M,$$for$$a_1a_2\cdots a_k = M.$$ But I can't understand that how $$\prod_{1\le i\le n}…
Silent
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a problem on optimization having a good looking

$$\min_{x\geq 0}\sum_{i=1}^n (a_i-x b_i)^2 [a_i-x b_i\leq 0],\quad a_i,b_i\in\mathbb R,n\in\mathbb N$$ where $[p]$ is an Iverson bracket. The objective function seemed easy (convex). 1.Is there any the concrete name for this kind of optimization or…
jerryren
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Calculating the quickest path between two points.

Determine the pattern according to the situation, the quickest route from checpoint 1 checkpoint to 2 when the orienteer's running speed on the stomping is 2.5 times as large as on the swamp. X=750m and Y=400m
typicalerror
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Lagrange method with inequality constraints

Please can someone help me with the follwoing optimisation: Minimize $-x+y$ subject to $0 \leq x \leq a$ and $0 \leq y \leq 1$ and $x^2 \leq y$. I am not sure how to deal with inequality constraints. Do I still make the Lagrangian:…
Natalie
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Prove that this is the solution to the given minimization problem

I have the minimization problem minimize $\displaystyle f_0 = \sum_{i=1}^{N} \mu_i \left( \left( 2^\frac{R_i}{\mu_i} - 1 \right) \right)$ with constraint $\displaystyle\sum_{i=1}^{N} \mu_i = 1$ and believe that the solution is $\displaystyle \mu_i…
Hauke
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Can we minimize $m\log_m{n}$, given $n$?

If we are given $n$, a positive real, can we find a the positive real $m$ that minimizes the function: $$m\log_m{n}$$ I'd prefer to find the function that gives a value for $m$, but I'm also interested in asymptotic bounds for $m$. This is similar…
Matt Groff
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minimizer of c/x

It seems to me that the solution to the optimization over $\lambda \in \mathbb{R}^n$ \begin{align} & \underset{\lambda}{\arg\min} \sum_i c_i/\lambda_i\\ \textbf{s.t } & \sum_i \lambda_i = 1\\ & \lambda_i \geq 0 \end{align} where $c \in…
fairidox
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(Proximal) subgradient inclusion property proof

I'm having a bit of trouble proving what seems to be two fairly straightforward statements for a nonlinear optimisation class I'm taking. We're studying properties of the proximal subgradient, $\partial_p(f) = \{ v\ |\ \exists \rho, \delta: f(y)…
Ben Stott
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How do I find the minimum positive integers for a set of 6 unknowns with the 3 inequalities?

I'm developing a turn-based RPG game that features three characters: Tank, Warrior, and Assassin. Each character has two attributes: maxHP and attack. Now, my goal is to create a balanced system where the Tank defeats the Warrior, the Warrior…
JJJohn
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How to optimize $F=(\sqrt{a^2-x^2}-y)^2+(c\sqrt{b^2-y^2}-x)^2, a,b,c>0$

$F=\left(\sqrt{a^2-x^2}-y\right)^2+\left(c\sqrt{b^2-y^2}-x\right)^2,\quad a,b,c>0$ When $c=1$, we can optimize $F$ by using the substitution $x=a \sin(u), y=b \cos(v)$, then $$F=a^2 \cos(u)^2+b^2 \cos(v)^2-2ab \cos(u)\cos(v)+b^2\sin(v)^2…
MathDona
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