Recent questions in Optimization

Multivariable calculusAnswered question

Brooke Richard 2022-11-09

A square and a equilateral triangle are to be formed out of the same piece of wire. The wire is 6 inches long. How do you maximize the total area the square and the triangle contain?

Multivariable calculusAnswered question

Adison Rogers 2022-11-04

How do you find the dimensions of a rectangle whose area is 100 square meters and whose perimeter is a minimum?

Multivariable calculusAnswered question

Stephany Wilkins 2022-10-27

How do you find two positive numbers whose sum is 300 and whose product is a maximum?

Multivariable calculusAnswered question

Paloma Sanford 2022-10-23

Let $V={R}^{d}$ and $({x}_{i},{y}_{i}{)}_{1\le i\le n}\in (V\times \{-1,1\}{)}^{n}$

Let $C=\{(w,b)\in V\times R:1-{y}_{i}({w}^{t}{x}_{i}-b)\le 0,\mathrm{\forall}i\in [1,n]\}$

Show that $mi{n}_{(w,b)\in C}||w|{|}^{2}$ has a solution $(w,b)$

What I tried:

$||w|{|}^{2}$ is clearly continuous, I proved that C is closed. If C is bounded, then C is compact and the solution exists (In this case, I don't know how to prove that C is bounded). If C is not bounded, I don't know how to conclude either.

Let $C=\{(w,b)\in V\times R:1-{y}_{i}({w}^{t}{x}_{i}-b)\le 0,\mathrm{\forall}i\in [1,n]\}$

Show that $mi{n}_{(w,b)\in C}||w|{|}^{2}$ has a solution $(w,b)$

What I tried:

$||w|{|}^{2}$ is clearly continuous, I proved that C is closed. If C is bounded, then C is compact and the solution exists (In this case, I don't know how to prove that C is bounded). If C is not bounded, I don't know how to conclude either.

Multivariable calculusAnswered question

Keyla Koch 2022-10-21

How do you find the largest possible area for a rectangle inscribed in a circle of radius 4?

Multivariable calculusAnswered question

Bairaxx 2022-10-18

A rectangular box is to be inscribed inside the ellipsoid $2{x}^{2}+{y}^{2}+4{z}^{2}=12$. How do you find the largest possible volume for the box?

Multivariable calculusAnswered question

Amina Richards 2022-10-13

How do you optimize $f(x,y)=2{x}^{2}+3{y}^{2}-4x-5$ subject to ${x}^{2}+{y}^{2}=81$?

Multivariable calculusAnswered question

Damon Vazquez 2022-10-06

How do you find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum?

Multivariable calculusAnswered question

Nathanael Perkins 2022-09-27

How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?

Multivariable calculusAnswered question

Vrbljanovwu 2022-09-27

What is the maximum possible area of the rectangle that is to be inscribed in a semicircle of radius 8?

Multivariable calculusAnswered question

Addyson Bright 2022-09-24

Consider a NLP $min\{f(x):g(x)\le 0\}$. There are no equality constraints. The problem is feasible for small steps $t>0$. I have to prove that $g(x+td)\le 0$ if $g(x)<0$, where $t$ is the step length and $d$ is the direction of the line search (gradient descent).

I was thinking that since $t$ is positive and the direction $d$ can not be negative (not too sure about this fact), hence their multiplication is positive. The only way for $g(x+td)$ to be 0 or negative is for $g(x)$ to be negative.

I was thinking that since $t$ is positive and the direction $d$ can not be negative (not too sure about this fact), hence their multiplication is positive. The only way for $g(x+td)$ to be 0 or negative is for $g(x)$ to be negative.

Multivariable calculusAnswered question

ct1a2n4k 2022-09-23

How do you find the length and width of a rectangle whose area is 900 square meters and whose perimeter is a minimum?

Multivariable calculusAnswered question

Thordiswl 2022-09-23

1- What is Optimization? How many methods are there to calculate it?

2- What do we mean by an objective function? What do we mean by constraints?

3- Give three practical examples (physical or engineering) of a target function with a constraint

2- What do we mean by an objective function? What do we mean by constraints?

3- Give three practical examples (physical or engineering) of a target function with a constraint

Multivariable calculusAnswered question

koraby2bc 2022-09-23

What are the radius, length and volume of the largest cylindrical package that may be sent using a parcel delivery service that will deliver a package only if the length plus the girth (distance around) does not exceed 108 inches?

Multivariable calculusAnswered question

Linda Peters 2022-09-20

The productivity of a company during the day is given by $Q\left(t\right)=-{t}^{3}+9{t}^{2}+12t$ at time t minutes after 8 o'clock in the morning. At what time is the company most productive?

Multivariable calculusAnswered question

hotonglamoz 2022-09-20

How do you find the points on the parabola $2x={y}^{2}$ that are closest to the point (3,0)?

Multivariable calculusAnswered question

Keenan Conway 2022-09-19

I have a knapsack problem

$\begin{array}{rl}& \underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}\\ & \text{s.t.}\sum _{i=1}^{n}{w}_{i}{x}_{i}\le c.\end{array}$

The Lagrangian relaxation is as follows

$\begin{array}{r}\underset{\lambda \ge 0}{min}\underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}-\lambda (\sum _{i=1}^{n}{w}_{i}{x}_{i}-c).\end{array}$

Suppose I solved the relaxed problem and got an optimal ${x}_{lag}$ s.t. $f({x}^{\ast})<f({x}_{lag})$ where ${x}^{\ast}$ is the optimal solution of the original problem and $f$ is the objective function. Even though ${x}_{lag}$ gives a strict bound, is it consideblack to be a good approximate solution?

Is it true that the relaxation can be solved in polynomial time since the dual problem is convex in $\lambda $ and the maximization part with fixed $\lambda $ is just activating ${x}_{i}$ associated with the largest term $({v}_{i}-\lambda {w}_{i})$?

$\begin{array}{rl}& \underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}\\ & \text{s.t.}\sum _{i=1}^{n}{w}_{i}{x}_{i}\le c.\end{array}$

The Lagrangian relaxation is as follows

$\begin{array}{r}\underset{\lambda \ge 0}{min}\underset{x\in \{0,1{\}}^{n}}{max}\sum _{i=1}^{n}{v}_{i}{x}_{i}-\lambda (\sum _{i=1}^{n}{w}_{i}{x}_{i}-c).\end{array}$

Suppose I solved the relaxed problem and got an optimal ${x}_{lag}$ s.t. $f({x}^{\ast})<f({x}_{lag})$ where ${x}^{\ast}$ is the optimal solution of the original problem and $f$ is the objective function. Even though ${x}_{lag}$ gives a strict bound, is it consideblack to be a good approximate solution?

Is it true that the relaxation can be solved in polynomial time since the dual problem is convex in $\lambda $ and the maximization part with fixed $\lambda $ is just activating ${x}_{i}$ associated with the largest term $({v}_{i}-\lambda {w}_{i})$?

Multivariable calculusAnswered question

unjulpild9b 2022-09-17

A cylindrical can is to be made to hold 1000cm^3 of oil. How do you find the dimensions that will minimize the cost of metal to manufacture the can?

Multivariable calculusAnswered question

overrated3245w 2022-09-07

What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 125 $c{m}^{3}$?

Optimization in calculus stands for the process where you aim to find the maximum and minimum values by the provided constraints. This is exactly where the calculus is implemented. You may either go with the equations or the word problem as most answers do. The trick is to find the maximum profit for something or the minimum time it will take to complete an operation. It’s what makes optimization in calculus so important. See several answers that are provided below to use the data as the template for your assignment. It will help you t find the most efficient solutions.