What is Optimization: Definition and 627 Discussions

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.

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  1. S

    Optimizing Cost of Half Cylinder Structure: 225K Vol

    Homework Statement - Building a half cylinder structure. - The structure must have an exact volume of 225,000 cubic feet. - The current construction costs for the foundation are $30 per square foot, the sides cost $20 per square foot, and the roofing costs $15 per square foot. - Minimize the...
  2. H

    Production Lot Size - Optimization long and difficult

    Homework Statement Non-Slip Tile Company has been using production runs of 100,000 tiles, 10 times per year, to meet the demand of 100,000 tiles annually. The set-up cost is $5,000 per run and (annual) holding cost is estimated at 10% of manufacturing cost of $1 per tile. Production capacity...
  3. U

    Modeling and Optimization

    We have just begun this topic and I'm really confused about how to approach questions, is there any trick or guideline for doing so? Ex: Consider an isosceles right triangle whose hypotenuse is the x-axis and whose vertex is on the y-axis. If the hypotenuse is 2 units long, we'd have...
  4. S

    Optimization of ellipsoid tube

    Homework Statement Problem 2 b) in the following link http://www.math.ubc.ca/~haber/courses/math253/Welcome_files/asgn4.pdf" Homework Equations V=pi(r1r2)H SA=? The Attempt at a Solution I was thinking I should form two equations V=10=pi(r1r2)h and then an equation for the...
  5. P

    Maximizing Volume, Area, and Rent: Optimization Problems Homework

    Homework Statement If you can help me answer ANY of these, it will be very appreciated. thanks in advance. 1. The following problem was stated and solved in the work Nova stereometria vinariorum, published in 1615 by the astronomer Johannes Kepler. What are the dimensions of the cylinder...
  6. A

    Optimization using differentiation

    Homework Statement The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $600 per person per day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 90) for the cruise, then each fare is reduced by $4...
  7. Telemachus

    Optimization: maximize a triangle surface

    See if anyone can help me with this: Among all triangles of perimeter equal to P, find the one with the largest area. (Hint: use the formula A=\sqrt[ ]{p(p-x)(p-y)(p-z)} where P=2p, P is the perimeter). So, I have f|_s , I think that must be solved using Lagrange multipliers, at least I don't...
  8. A

    Optimization using differentiation

    Homework Statement http://i53.tinypic.com/1zu5ty.jpg The Attempt at a Solution well so far, all i got is 3x + y = 3000; also y = 1000/x ==>> 3x+ (1000/x) = 3000 i don't know what the area should be though... would it be A=x2y If i am right in that, would i do this after i...
  9. E

    Least-square optimization of a complex function

    Dear all, I have a least square optimization problem stated as below \xi(z_1, z_2) = \sum_{i=1}^{M} ||r(z_1, z_2)||^2 where \xi denotes the cost function and r denotes the residual and is a complex function of z_1, z_2. My question is around ||\cdot||. Many textbooks only deal with...
  10. P

    Maximizing Intercepted Lengths in a Right Triangle Inscribed in a Circle

    Homework Statement A right angle is moved along the diameter of a circle of radius a (see diagram). What is the greatest possible length (A+B) intercepted on it by the circle. Homework Equations so, the pythagorean theorem might be useful diameter = 2a The Attempt at a Solution...
  11. P

    Optimize Function: Abs Max & Min Values of -2x^2 + 3x + 6x^(2/3) + 2

    Homework Statement Find the abs max and abs min values of the function f(x) = -2x^2 + 3x + 6x^(2/3) + 2. Homework Equations The Attempt at a Solution So the candidates are the endpoints, where f'(x) = 0, and where f(x) DNE. f(-1) = 3 f(3) = 5.481 For the derivative of...
  12. T

    Differential Calculus: Solving for Optimization

    1. Hey all, For my calculus class we were giving the problem of solving for the optimization of a tin can using differential calculus. The problem was to find the minimum cost for any tin can of any height(as well as using the equation for the tin we had). The surface area of the cylinder was...
  13. K

    Constrained Optimization via Lagrange Multipliers

    Hi, I'm trying to do a constrained optimization problem. I shall omit the details as I don't think they're important to my issue. Let f:\mathbb R^n \to \mathbb R and c:\mathbb R^n \to \mathbb R^+\cup\{0\} be differentiable functions, where \mathbb R^+ = \left\{ x \in \mathbb R : x> 0...
  14. P

    New logic optimization algoritm

    hi I have an idea for new logic optimization algoritm, like "Quine–McCluskey algorithm" and the "Espresso heuristic logic minimizer", but it can handle multi-level representations and it can find the (theoretical) best circuit. It should work for 8 to 12 input bits. I was wondering if such...
  15. A

    Optimization (min/max and concavity)

    This isn't a homework question, although I am in a calculus course. I'm a little fuzzy on the method that I was taught (discover intervals and all that nonsense to make sure f'(x)=0 is a max or a min). I was curious if, when I discovered the values of x such f'(x)=0, I could then find f''(x)=0...
  16. M

    Finding the derivative in an optimization problem

    Hi all, I've been stuck on this question for hours and hours, I'm not sure what I'm doing wrong.. The question states, "a new cottage is built across the river and 300 m downstream from the nearest telephone relay station. The river is 120 m wide. to wire the cottage for phone service, wire...
  17. S

    Implicit differentiation and optimization

    Homework Statement A conical tent must contain 40\pi ft^{3}. Compute the height and radius of the tent with minimal total surface area. (Include the floor material.) Homework Equations 1. \frac{\pi r^{2} h}{3} = 40\pi 2. \pi r \sqrt{r^{2} + h^{2}} + \pi r^{2} = S 3. \frac {dr}{dh} =...
  18. A

    Gaussian frequency job after optimization

    Hi, I already got some good Gaussian-help in this forum, so maybe somebody can help me once again :) I did an optimization run for my structure in Gaussian and didn't know, that I could have combined this with a frequency calculation. So now I have to start a new frequency job based on my...
  19. V

    Multivariable Constrained Optimization

    hi i want to find values of a,b,c such that.. Minimize (a+b+c) constrained to (x-a)^2 + (y-b)^2 + (z-c)^2 less than equal to R(z) (x-a)^2 + (y-b)^2 + (z-c)^2 greater than equal to r(z) can anyone help me solving this?? which method should b used for better computation??
  20. A

    Optimizing Dimensions and Cost in Golf Net and Fencing Projects

    Homework Statement Two related type of questions: 1) A rectangular prismic net enclosure for practising golf shots is open at one end. Find the dimensions that will minimize the amount of netting needed and give a volume of 144 m3. Netting is only required on the sides, top, and the far...
  21. R

    Complex Optimization Problem regarding Areas

    Homework Statement Part 1: A forest in the shape of a 50km x 50 km square has firebreaks in rectangular strips 50km by 0.01 km. The trees between two fire breaks are called a stand of trees. All firebreaks in this forest are parallel to each other and to one edge of the forest, with the first...
  22. O

    What is the Cheapest Program to Train a Senior Manager with Specific Skills?

    Goal: You want to train your Senior Manager. He needs skills: x1, x2, x3, x4, x5, x6, x7, x8, x9, x10. You are to choose from the following programs/courses that fulfills all the senior manager's skill needs at the cheapest cost. p1 has x1, x3, x4 at $500 p2 has x3, x5, x9, x10 at $1000...
  23. L

    Geometry Optimization with GAUSSIAN 03W

    Hello, I am new in computational chemistry. I was calculating by "DFT and HF theory" (using GAUSSIAN 03W) molecular parameters of "2D coordination polymer, [Cd(μ-pydc)(2-mim)]n (pydc = pyridine-2,3-dicarboxylate, 2-mim = 2-methylimidazole)" . I have Crystallographic data are belong to this...
  24. F

    Optimization of C code: smoothing an image.

    Homework Statement I need to optimize this given code: /* A struct used to compute averaged pixel value */ typedef struct { int red; int green; int blue; int num; } pixel_sum; /* Compute min and max of two integers, respectively */ static int min(int a, int b) { return (a < b ...
  25. T

    Proving Maximum Volume of a Right Circular Cone: Optimization Problem Solution

    Homework Statement A right circular cone of base radius r and height h has a total surface area S and volume V . Show that 9V2=r2(S2-2pir2S) . (i can do this part) . Hence or otherwise , show that for a fixed surface area S , the maximum volume of the cone occurs when its semi-vertical angle...
  26. F

    Optimization of C Code Loop Unrolling

    Homework Statement I need to optimize this given code that rotates an image 90 degrees so it runs at least three times faster: void naive_rotate(int dim, pixel *src, pixel *dst) { int i, j; for (i = 0; i < dim; i++) for (j = 0; j < dim; j++) dst[RIDX(dim-1-j, i, dim)] =...
  27. D

    Maximize Volume of Trough: Find Theta Value

    The problem states: The trough in the figure is to be made to the dimensions shown. Only the angle theta can be varied. What value of theta will maximize the troughs volume? http://img81.imageshack.us/img81/5963/24ni3.jpg (There is an image of the problem) I know the height in terms...
  28. Z

    I am struggling so much with optimization.

    Is anyone able to give me some pointers. I am trying to brush up on my calculus I for my next calc class and I can't grasp optimization. I hated it then and I hate it now. I can do everything else in calculus I except this and it's so irritating. I can learn every integration rule in Calc I in a...
  29. E

    Maximizing XY: Problem 2 Solution - No Numbers Given

    1.http://www.teachingcenter.ufl.edu/materials/math_lab/oldtests/FA09_MAC2311_exam4ab.pdf Number 2 2. Maxamize XY subject to , y=sqrt(x) 3. I don't know what numbers to use...
  30. S

    Numerical Optimization ( norm minim)

    Homework Statement Consider the half space defined by H = {x ∈ IRn | aT x +alpha ≥ 0} where a ∈ IRn and alpha ∈ IR are given. Formulate and solve the optimization problem for finding the point x in H that has the smallest Euclidean norm. Homework Equations The Attempt at a...
  31. H

    Rectangular container optimization

    Homework Statement A rectangular storage container with an open top is to have a volume of 10 m^3. The length of this base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such...
  32. S

    Numerical Optimization ( steepest descent method)

    Homework Statement Consider the steepest descent method with exact line searches applied to the convex quadratic function f(x) = 1/2 xT Qx − bT x, ( T stands for transpose). show that if the initial point is such that x0 − x* ( x* is the exact solution of Qx = b) is parallel to an...
  33. M

    Maximize Volume of Cone: Homework Equations & Solution

    Homework Statement Homework Equations Volume of cone= (1/3)*pi*r^2*h Volume of sphere= (4/3)*pi*r^3 Surface area of sphere 4*pi*r^2 The Attempt at a Solution primary equation is V(cone)= (1/3)pi*r^2*h---> V(cone)= (1/3)pi*(r-h/2)^2*h constraint: constraint:V(sphere)= (4/3)*pi*r^3 ***from...
  34. R

    Optimization - Minimizing the cost of making a cyclindrical can

    Homework Statement The can will hold 280 mL of juice. The metal for the side of the can costs $0.75/m^2. The metal for the top and bottom costs $1.4/m^2. The side of the can is one rectangular sheet. The top and bottom are stamped out from another rectangular sheet, the unused metal from this...
  35. C

    Optimization of a rectangle's area in two parabolas

    Homework Statement Determine the maximum area of a rectangle formed in the region formed by the two curves y1=x2 - k y2=x2 + k Homework Equations The equations are given, I tried using k=1. so y1= x2 - 1, etc. The Attempt at a Solution Is it true that the rectangle has to be...
  36. R

    Optimization - Volume of a Box

    Homework Statement Ok I know this question is really easy but for some reason I got it wrong. You are given a piece of sheet metal that is twice as long as it is wide and has an area of 800m^2. Find the dimensions of the rectangular box that would contain a maximum volume if it were...
  37. R

    Optimization - gothic window

    Homework Statement a gothic window it to be built with 6 segments that total 6m in length. The window must fit inside an area that is 1m wide and 3 meters tall. the triangle on top must be equilateral. What is the maximum area of the window. Homework Equations The Attempt at a...
  38. A

    How Do You Determine the Optimal Station Location Between Two Towns?

    Hi, I am having a hard time with this Optimization question as i do not know where to begin, I drew a diagram but what formulas, function etc do I use to start the question? And How do i do it? Two towns A and B are 7km and 5km, respectively, from a railroad line. The points C and D nearest to...
  39. H

    Minimize Cost: Optimize Cost of Telephone Line Across River

    Homework Statement A telephone company has to run a line from point A on one side of a river to another point B that is on the other side, 5km down from the point opposite A. The river is uniformly 12 km wide. The company can run the line along the shoreline to a point C and then under the...
  40. C

    Optimization (applications of Differentation) Problem

    Homework Statement The demand function for a product is modeled by p=56e^-0.000012x Where p is the price per unit (in dollars) and x is the number of units. What price will yield a maximum revenue. Homework Equations The Attempt at a Solution Ok so i tried taking the...
  41. P

    How Much Should Rent Be Charged to Maximize Profit?

    Homework Statement A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. For ever $25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month. How much rent...
  42. P

    Optimization Problem - Calculus

    Homework Statement A cylindrical shaped tin can must have a volume of 1000cm3. Find the dimensions that require the minimum amount of tin for the can (Assume no waste material). The smallest can has a diameter of 6cm and a height of 4 cm. Homework Equations V = \pi r^{2}h P = 2(...
  43. C

    Optimization calculus problem

    Homework Statement The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 feet apart, where...
  44. L

    Optimization - area of rectangle

    Homework Statement If the perimeter of a rectangle is fixed in length, show that the area of the rectangle is greatest when it is square Homework Equations The Attempt at a Solution if the perimeter is fixed in length, then 2x + 2y = c then no idea to continue from there
  45. T

    Optimization - find two points on a curve with a common tangent line?

    Homework Statement Find two points on curve y=x4-2x2-x that have a common tangent line. Homework Equations *the one stated above dy/dx = 4x3-4x-1 The Attempt at a Solution equation of a tangent line: y=mx+b (4x3-4x-1) = m at two different points? So there are two points for which...
  46. T

    Optimization - help needed setting up a system of equations?

    Homework Statement A truck driving over a flat interstate at a constant rate of 50 mph gets 4 miles to the gallon. Fuel costs $0.89 per gallon. For each mile per hour increase in speed, the truck loses a tenth of a mile per gallon of its mileage. Drivers get $27.50 per hour in wages, and the...
  47. L

    Optimize Derivative of Trig Functions Grade 11 Math

    Homework Statement there's a picture of the question... from my textbook http://photos-h.ak.fbcdn.net/hphotos..._1385551_n.jpg thers a diagram image of the problem too to help understand Homework Equations well its a word problem, i used cosine rule at beginining and then...
  48. L

    Derivative optimization trig functions, give it a try please grade 11 math

    could someone please try and solve this? and explanation would be greatly appreciated too ! this was one of the homework questions, but i didnt really understand. the teacher explained it again to the class partly, but didnt understand a part of it so we didnt continue... maybe one of you guys...
  49. L

    Optimization of inscribed circle

    Homework Statement Given the function y = 12- 3x^2, find the maximum semi-circular area bounded by the curve and the x-axis. Homework Equations A= Pi(r^2) The Attempt at a Solution I found my zeros, 2 and -2, and my maximum height of 12 from the y'. A' = 2Pi(r)
  50. D

    Sedumi for semidefinite optimization

    Dear Friends, Some one can help me to make sedumi function which can solve the semidefinite optimization. Additionally, how can I distinguish between Second order cone programming and semidefinite quadratic linear programming. Thank You.
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