Mathematics Stack Exchange
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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").

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Optimizing Measurements - Grade 9

You need to ship USB cables in a small cardboard square-based prism box. The box must have a capacity of 750 cm3 but use the smallest amount of cardboard. a) What are the dimensions of the box? (2 decimal places) b) How much cardboard will you be…
Chelsea
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Optimization of product of standard normal CDFs.

I want to maximize $\phi(c_{1},c_{2}) = \Phi(c_{1})\Phi(c_{2})$ given $c_{1}+c_{2} = 1$, where $\Phi(\cdot)$ is the CDF of standard normal distribution and $0\leq c_{i}\leq 1$ for $i=1,2$. I did it using simulations and found the solution at $c_{1}…
Satya Prakash
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Analytic solution to this?

This came up as the solution to a constrained optimization problem (after doing some analysis to find the stationary points of some Lagrangian): $$ A_{ij} \exp(\eta_j) = B_{ij} \exp(\eta_i) $$ $A$,$B$ are square of dimension $n\times n$. $B$ is…
tangerine
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Minimize (xy-1)^2 + x^2

I am practicing for my final exam and I came across this exercise online. Yet, I have no clue how to solve it. min (xy-1)^2 + x^2 I figured the four step method would be a good idea. I got stuck on how to prove Weierstrass. Also, the solution I…
Anonymous
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What is the optimization metric if the goal is to allocate each person the fewest number of types given several types of items.

It is really hard for me to describe my optimization goal. If anyone finds a better title, please just replace it. Here is an example of my optimization problem. We have 3 people: A,B,C, and we have 5 apples, 5 bananas, and 3 pears. The constraint…
MIMIGA
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Maximization using the first derivative, volume of a cone

A huge conical tank to be made from a circular piece of sheet metal of radius 10m by cutting out a sector with vertex angle theta and welding the straight edges of the straight edges of the remaining piece together. Find theta so that the resulting…
jw35174
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optimality conditions

there is a Theorem If x is a local optimum to (G) then there is no feasible descent direction at x. (G) min f(x) I want to know if the reverse of theorem is true or not, that means if there is no feasible descent direction at x, then x is local…
user509069
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inventory optimization algorithm

I have been researching different ways to optimize inventory and came across this text by, "Nuri Sercan Ozbay" which states, Does anyone know how to actually apply this algorithm? Or has anyone actually seen this before?
Alex Fields
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What are the best optimization algorithms for smooth function with vast flat areas?

I have a set of smooth functions that I need to find their maximum (not necessarily the global one, the local ones are fine too). Here are some characteristics of the mentioned functions: Not known beforehand, they are generated by a program and…
Sachamora
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Maximization problem subject to sum of square restriction

Let $(c_1,c_2)$ be a fixed point in $\mathbb{R}^2$. How to maximize $|x_1x_2-c_1c_2|$ subject to the condition that $(x_1-c_1)^2+(x_2-c_2)^2<1$. i.e. $$\sup_{(x_1,x_2) \in \mathbb{R}^2:(x_1-c_1)^2+(x_2-c_2)^2<1} |x_1x_2-c_1c_2|=?$$ Note : This is a…
user469680
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Can I solve $\min_{x,y}f(x)g(y)\;s.t.x\in\mathcal{X},y\in\mathcal{Y}$ by solving its equivalent problem?

Consider the following problem $$\min_{x,y}f(x)g(y)\\ \text{s.t. } x\in\mathcal{X},y\in\mathcal{Y}$$ where $\mathcal{X},\mathcal{Y}$ are two convex sets, $f(x),g(y)>0$. If I want to solve this problem, can I solve the following problem,…
Dave
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Global minimum for a function on an unrestricted domain

We have a function $f(x,y) = (\dfrac{3}{2} - x + xy)^2 + (\dfrac{9}{4} - x + xy^2)^2$ I know that $\nabla f(0,1) = \nabla f(3,0.5) = 0$ and that $f(3, 0.5) < f(0,1)$. I am wondering whether or not this is sufficient to decide that the global minimum…
YakSal Tafri
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A Minimization Problem Involving a Summation

Please help me with the following problem: $$\text{minimize }\sum\limits_{i=1}^{t}{{{m}_{i}}.\left( \frac{{{m}_{i}}-1}{2} \right)},$$ subjected to: $$\sum\limits_{i=1}^{t}{{{m}_{i}}=n-1},$$ and $$t<\frac{n}{2},$$ where ${{m}_{i}}$ and $t$ are…
Mehrdad
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Optimize y coordinates of dots for minimally overlapping connections

I have a chart with a variable amount of dots. All dots are connected to one or more other dots via a line. The x values of these dots are fixed, but the y-values may vary. How do I get the optimal configuration of y-values for these dots? A simple…
Michiel Borkent
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Maximum and Minimum Value of function

I need help on finding the maximum and minimum value of the function $g$. I found the stationary points of g function by putting the first order conditions $=0$. I got $y=0$ and $x=z$. Also, the maximim value of f function is $f(0,0,0)=9.$ To find…
Bunty
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