Optimization Definition and 629 Threads

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. I

    How to Solve Optimization Problems with Multiple Variables and Constraints?

    Well, I'm having trouble doing optimization problems (maximizing and/or minimizing a function in more then one variable with/without constraints). Would be a great help if someone could give me some good links on this topic or some methods generally. If the domain is compact; where are the...
  2. I

    Optimization problems involving non-compact domains

    I have some understanding of how to solve problems involving compact domains. Set the gradient to zero and solve for x and y, and then try to parameterize if needed to find max/min over the border of the domain. The thing is, my book doesn't go into much detail on how to do optimize functions...
  3. M

    Optimization Solver - BFGS method with bound constraints

    Hello, I am working on a research project that requires me to write a solver for solving a particular problem. I could really use some math advice if anyone is willing to assist. I need to minimize a non-linear objective functions of 5 variables. It is a pretty complex function. Each of the...
  4. M

    Is this optimization problem missing information?

    Homework Statement A 5324 cubic foot tank with square base and an open top is to be constructed of a sheet of steel of a given thickness. Find the length of a side of the square base of the tank with minimum weight.Homework Equations The Attempt at a Solution I'm usually fairly decent at...
  5. D

    Optimization problem involving an area and perimeter

    My friend and I have come across this problem in Apostol's Calculus Vol. 1, ed.2 (exercises 4.21 if anyone is looking). We are studying calculus independently and have become stumped by this one. Homework Statement The problem as written in Apostol: "A farmer wishes to enclose a rectangular...
  6. U

    Optimization of a fence around a triangular pen

    Homework Statement A farmer wishes to enclose a pen in the shape of a right triangle with 100 ft of fencing. Set up the equation to find the maximum and minimum dimensions but do not solve the problem. Homework Equations I know the area for a triangle is simply A=1/2B*H and that the...
  7. S

    Help with really hard optimization problem

    Homework Statement A construction company has been offered a contract for $7.8 million to construct and operate a trucking route for five years to transport ore from a mine site to a smelter. The smelter is located on a major highway, and the mine is 3 km into a heavily forested area off...
  8. B

    Optimization problem using exact Hessian

    hi, I'm kind of new to optimization theory, and I have to maximize a multi-dimensional problem where I know the exact gradient and hessian. In other words, techniques such as BFGS are not sufficient because I don't want to approximate the Hessian (with an initial guess for example of H=I), I...
  9. QuarkCharmer

    Optimization Problem Homework Solution

    Homework Statement I took a test today on integration, curve sketching, and optimization. I am pretty sure that I got a 100 on it due to all the help here on PF with indef. integration and all of the helpful u-sub advice I have received. Anyway, there was 5 optimization word problems, and...
  10. QuarkCharmer

    Optimization of box, varied material cost.

    Homework Statement Stewart Calculus 6E: 4.7 #14 A rectangular storage container with an open top is to have a volume of 10m³. The length of it's base is twice the width. Material for the base costs $10 per square meter. Material for the sides cost $6 per square meter. Find the cost of...
  11. QuarkCharmer

    How Can You Optimize Costs and Profits in These Mathematical Problems?

    Homework Statement #56.) Someone makes necklaces and sells them for 10 dollars each. His average sales were 20 per day. When he raises the price to 11 dollars per day, the average sales drops 2. a.)Find the demand function, assuming it is linear. b.)If the material to make each necklace...
  12. M

    Optimization problem/finding domain and proof

    Homework Statement a metal box with square base a no top holds 1000 cubic centimeters. it is formed by folding up the sides of the flattened pattern picture and seaming up the four sides. the material for the box costs $1.00 per square meter and the cost to seam the sides is 5 cents per meter...
  13. J

    Optimizing Area in a Semi-Circle

    1. Find the dimensions of the rectangle with the largest area that can be inscribed in the upper semi-circle given by x^2+y^2 ≤ 16, y≥0. 2. I thought I'd use A=lw 3. This is but a guess..so take it with a grain of salt.. height=2x base= x^2+y^2 A(x) = 2x(x^2+y^2) = 2x^3+2xy^2 A'(x) =...
  14. B

    Mathematica Hessian optimization through Mathematica

    I know by default that Mathematica will use the BFGS method when you request "FindMinimum[Function]" but I am curious for a hint towards a pseudo-code for the following problem: I have a collection of functions, say F = {f1,f2,...,fN} and I want to transform them as linear combinations of one...
  15. S

    HELP calculus optimization problem: fitting thin rod through corridor

    I am having trouble conceptualizing a calculus optimization problem. I can find the answer to the problem by using the procedure but i am quite uncertain of how the equations match up with what's actually going on in the situation! Problem: What is the max length of widthless rigid pole that...
  16. Femme_physics

    Optimization, finding two numbers whose sum is minimal

    Homework Statement From all positive numbers x and y that hold y(x+2) = 9 , find the two numbers whose sum x+y is minimal The Attempt at a Solution Attached. My idea here is to take the derivative of y with respect to x, and set it equal to zero. This is how I understand you solve...
  17. H

    Optimizing Rocket Equation Mass Ratios with MATLAB fmincon

    Homework Statement I'm trying to use fmincon to find the optimal mass ratios for the rocket equation. The only variables are m2/m1 and m3/m2 (R1 and R2 respectively). The Attempt at a Solution function [ velocity ] = rocketEqn( MR ) % Calculates the velocity that a rocket can attain...
  18. L

    How can I optimize the area of a roof with a given wall length and angle?

    Homework Statement [PLAIN]http://img593.imageshack.us/img593/7536/unledci.png Homework Equations The Attempt at a Solution I called the wall b, half the roof a, and the angle theta. I get Area=2ab*sin(theta/2)+1/2 a^2 sin(theta)... try differentiating with respect to a and...
  19. H

    Difficulty with optimization problems

    I'm going through a calculus textbook in an attempt to learn it myself. So far so good, but I've been stuck on optimization problems. I understand the concept. The maxima and minima of a function can be found by looking at where its derivative = 0. I also see that a function that has no...
  20. A

    Constrained Least Square Optimization

    Hi, I want to know the solution of the following equation. a = argmin_{a}[\sum{||a^Tx_i - y_i||^2}+\alpha ||a||^2] \\ where x_i, y_i are column vectors of dimensions m and n respectively where m>n. \alpha is a scalar and Y = a^T X where X=[x_1 x_2 ... x_k], Y = [y_1 y_2 ... y_k] I...
  21. C

    Optimizing Local Business Profit: Calculus AB Project

    Homework Statement Hey everyone, I am in Calculus AB in high school and my teacher has given us a project to optimize a realistic situation. We are suppose to model a local business, but any type of business that uses optimization will suffice. We are suppose to create a profit function...
  22. S

    Stock market optimization fantasy

    Would the optimal trading strategy for this stockmarket optimization fantasy be trivial or nearly impossible to compute? -or something in between? You have an initial amount of money A_o and your goal is to maximize the amount of money you will have at the end of a year by trading stocks...
  23. M

    Optimization question - optimal conical container

    Design the optimal conical container that has a cover and has walls of negligible thickness. The container is to hold 0.5 m^3. Design it so that the areas of its base and sides are minimized. information : 1) areas of the sides = (pi) x r x s 2) areas of the base = (pi) x (r^2) 3)volume of...
  24. E

    OPTIMIZATION: Minimizing Packaging Costs

    Homework Statement A rectangular box is to have a square base and a volume of 20 ft3. If the material for the base costs $0.30/square foot, the material for the sides costs $0.10/square foot, and the material for the top costs $0.20/square foot, determine the dimensions of the box that can be...
  25. A

    Optimization in economics help

    Homework Statement A grocer usually buys 30 dozen pineapples per week. The cost per pinapple is $0.40 and they are sold at 0.78 a piece. As there is an abundant crop on May, the wholesaler offers a discount of $0.05 per pineapple for each additional order of 5 dozens (i.e. if the grocer buys 35...
  26. E

    Optimizing Volume of a Rectangle with Newton's Method?

    Hi, If we have a problem with two n-dimensional vectors, can we still apply Newton method to find the minimum point? Regards
  27. L

    The best method for a many many variable optimization problem?

    I need to optimize a maximum likelihood function with many many variables (~10^2 variables). what is the faster method? thanx
  28. S

    Optimization of a suspended system. (hanging mass)

    Homework Statement A load must be suspended 6m below a high ceiling using cables attached to two supports that are 2m apart. How fare below the ceiling (x in figure) should the cables be joined to minimize the total length of the cable used? They give a figure, which I am butchering here...
  29. E

    What is the Most Efficient Optimization Algorithm?

    Hi, I have a problem to solve using a sequential optimization algorithm. But since there are many algorithms, I am now confused which one to use. Which one is the most efficient? Thanks
  30. R

    What is the Objective Function and Constraints for this Optimization Problem?

    Homework Statement Optimization (Maximize or Minimize) JJCJ=-x +2y according to: A(1,2) B(-1,2) C(-1,-3) Homework Equations The Attempt at a Solution I have taken many advanced math courses and its kind of embarrassing that I don't know how to approach this question :\...
  31. L

    Optimization Problem Homework: Find Largest Positive Number

    Homework Statement Find the positive number that exceeds its square by the largest amount. Obviously this is on the open interval (0,1). Homework Equations The Attempt at a Solution F(x) = ( \frac{1}{n} ) ^2 - n \Rightarrow F'(x) = \frac{-2}{n^3} - 1 = 0 \Rightarrow 1 =...
  32. M

    What are the dimensions of the cedar chest that minimize the cost?

    Homework Statement The length of a cedar chest is twice its width. The cost/dm^2 of the lid is four times the cost/dm^2 of the rest of the cedar chest. If the volume of the cedar chest is 1440 dm^3, find the dimensions so that the cost is a minimum. Homework Equations LWH = 1440 W = 2L...
  33. D

    How can I find the optimal vector x for a constrained optimization problem?

    Hi all, I am working on a project and stuck at the following problem. Find vector x_{n\times 1} which minimizes the function f(x) = \sum_{i}^{n}x_{i}^{2} subject to the linear equality constraint [A]_{m\times n} x_{n \times 1}=b_{m\times 1} with m\leq n The function f(x) trivially...
  34. R

    I need a good book for Linear and Non Linear Optimization

    i need a decent book for linear and non linear optimization. Currently i am using Linear and Non linear optimization by Griva Nash and Sofer, and it is by far the worst math book i have ever used. It does not have any solved examples or anything. It does not even have any proofs. It has...
  35. R

    Optimization problem, local minima and feasible set

    Homework Statement minimiza f(x) = x_1 subject to (x-1)^2+y^2=1 (x+1)^2+y^2=1 Graph the feasible set, Are there any local minimizers and global minimizers? Homework Equations I have graphed the feasible set...
  36. A

    What is the optimal amount of wire to use for a circle to minimize its area?

    Homework Statement A piece of wire 12 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (Give your answers correct to two decimal places.) Part A) how much of the wire should be used for the circle to maximize the area? (Solved this part...
  37. A

    Optimization find dimensions problem

    Homework Statement The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the smallest area. Homework Equations Area of a rectangle is length...
  38. R

    Optimization proof for Ax > b. Prove that set is convex

    Homework Statement Consider a feasible region S defined by a set of linear constraints S = {x:Ax<b} Prove that S is convex Homework Equations All what i know is that, a set is convex if and only if the elements x, and y of S ax + (1-a)y belongs to S for all 0 <a < 1 The...
  39. I_am_learning

    Turns / Core size optimization for Transformer

    Say, I want to design, 220 / 12 V 100VA transformer. We have V = 4.4BfNA (V is applied voltage RMS, B is peak flux Density, N is no. of turns, f is frequency, A is core cross section) so, B = V / (4.4 f NA) If i use iron core, there is limit to the maximum value of B without excessive...
  40. D

    Single variable optimization problem

    Homework Statement Jane is 2 miles offshore in a boat and wishes to reach a coastal village 6 miles down a straight shoreline from the point nearest the boat. She can row her boat at 5 mph and can walk at 3 mph. Where should she land her boat to reach the village in the least amount of time...
  41. S

    Maximizing Triangle Area with Given Adjacent Sides

    Homework Statement A triangle has adjacent sides 4 cm and 6 cm. Find the angle contained by the sides which maximizes the area. Homework Equations The Attempt at a Solution I'm not going to lie. I have no idea how to start this. I tried using sine law to create a helper equation...
  42. O

    Why Does Maximizing a Function Also Maximize Its Logarithm?

    the value same which maximizes the logarithm of the function and the plain form of the function why?? please help me, thanks
  43. D

    Optimization under differentiation

    Optimization under differentiation! Homework Statement OK I have a upside down looking curve structure (½ ellipse). It has the following specifications: The building has a rectangular base 150m long and 72m wide. The max height of the structure should not exceed 75% of its width or be less...
  44. S

    Optimization maximum area Problem

    1) The question A rectangular pen is to be built with 1200 m of fencing. The pen is to be divided into three parts using two parallel partitions. A) Find the maximum possible area of the pen. (45000 m^2) B) explain how the maximum area would change if each side of the pen had to be at least...
  45. A

    Optimization ( Applied Max and Minimum )

    Homework Statement From a square piece of cardboard, 30 cm on each side, an open topped box is to be constructed by cutting the squares from the corners and turning up the sides. What are the dimensions of the box of largest volume? The Attempt at a Solution I know how to do...
  46. R

    Optimization problem, triangle

    Homework Statement a line passes through the point (1,1/8) and intersects the positive x-axis at the point A and the positive y-axis at the point B. What is the shortest possible distance between A and B? Homework Equations i came up with three slopes for this line m1=-b/a...
  47. C

    Optimization word problem - minimizing surface area to find least expensive tank

    Homework Statement A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank covers $10 per square meter for the base and $5 per square meter for the sides, what...
  48. M

    Optimization, possibly just algebra help

    Homework Statement Find the critical points of the function. Then use the second derivative test to determine whether they are local minima or maxima(or state that the test fails). f(x,y)=(x-y)(e(x2-y2)) The Attempt at a Solution fx=(x-y)(2x(e(x2-y2)))+(e(x2-y2))=0...
  49. T

    Optimizing Volume of Inscribed Cylinder in Cone

    Homework Statement A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder. Homework Equations Vcone = (1/3)(pi)(r2)(h) Vcylinder = (pi)(r2)(h) The Attempt at a Solution I've been trying to relate the...
  50. F

    Optimizing Disk Submersion Height for Maximum Wetted Area

    Homework Statement A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquids as to maximize the exposed wetted area of the disk, show that the center of the disk should be positioned at a height r/ \sqrt{1+\pi^2}...
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