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. T

    Optimization program using Lagrange multipliers.

    Homework Statement Here is the problem, the solution and my question (in red): I'm guessing it was rejected because for the volume function, the dimensions cannot be negative? What if it was not volume and instead was just an arbitrary function. In that case you would not reject...
  2. MarkFL

    MHB Travis Henderson's Question: Optimizing f(x,y,z) with Constraint

    Here is the question: Here is a link to the question: Find the maximum and minimum values of f(x,y,z)=x^4+y^4+z^4 subject to the constraint x^2+y^2+z^2=1.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  3. D

    How Does Particle Swarm Optimization Enhance Power System State Estimation?

    I'm a final year student. My final project is about power system state estimation by using particle swarm optimization (PSO). I need to create a software based on PSO. Can somebody give me some brief idea on how to start the coding and guide me about the PSO thing.
  4. I

    MHB An optimization problem with Newton's method

    Apply Newton's method to $f(x)=(x-2)^4+(x-2)^5$ with initial guess $x_0=3$. We can observe that the sequence converges linearly with rate constant $3/4$. Now apply the iterative mathod $x_{k+1}=x_k-4f(x_k)/f'(x_k)$. This method should converge more rapidly for this problem. But how to prove that...
  5. G

    Discrete Optimization Question

    Homework Statement 9. This is a simplified inventory problem.Suppose that it costs c dollars to stock an item and that the item sells for s dollars. Suppose that the number of items that will be asked for by customers is a random variable with the frequency function p(k). Find a rule for...
  6. S

    Optimization Problem(Linear Programming Model)

    Homework Statement A factory has stocked a lot of pipes (sufficient). Each standard pipe is 5-meter long. But this kind of pipe cannot be used directly. We should cut them into three types: 140cm, 95cm and 65cm. In addition, the proportion of these three types of pipes must be 2:4:1. In...
  7. S

    An optimization problem worthy of our attention

    Perhaps forum members can advance science by solving this optimization problem from The Protein Engineer http://proteneer.com/blog/?p=1557 My statement of it: Let M be a N x N symmetric matrix such that N is divisible by 3, all the diagonal entries are 0 and each other entry is either 0...
  8. M

    How Can I Determine the Minimum Volume of a Cube Given the Surface Area?

    Suppose you are given a problem to find the dimensions for the maximum volume of a cube given the surface area. These problems involve 2 equations, taking the derivative and setting it equal to zero (local minimum or maximum) and substituting the 2nd equation to find the parameters. However...
  9. A

    Calculus optimization problem?

    The illumination from a bulb varies directly as the intensity of the light and Intensity varies inversely as the square of the distance from the source. Two bulbs are placed 54 feet apart. The intensity, Ia, of bulb A is 64cd, and the intensity, Ib, of bulb B is 125cd. At how many feet from bulb...
  10. N

    Particle Swarm Optimization vs. Newton's Method

    I have been reading Stephen Boyd's book Convex Optimization and I have learned to form various problems like LP, QP, QCQP, SOCP or SDPs. I also learned about formulating SVM for classification problem as optimization problem. Now I am reading about Gradient Methods, Newton's method, etc...
  11. G

    MHB Possible title: Linear Optimization Problem: Finding Optimal Solutions

    Linear Optimization Problem follow up Maximize: z=2x2+5x2+x3 x1+x2+x3 less then or equal to 12 x1-x2 less then or equal to 15 x2+2x3 less then or equal to 10 x1, x2 and x3 is greater then or equal to 0 x1= x2= x3= s1= s2= s3= z= I get x1=2, x2=10 x3=0 s1=0 s2=0...
  12. D

    Optimization of a rectangular window surmounted on a semicircle

    Homework Statement A decorative window has the form of a rectangle surmounted by a semicircle whose diameter is equal to the top of the rectangle. If the TOTAL perimeter of the window 16+pi, then what is the maximum area? A. 25.653 B. 32.148 C. 15.923 D. 38.047 E. 30.018 Correct...
  13. D

    Optimizing Profit for Wholesale Paint Dealer

    Homework Statement A wholesale paint dealer is buying and distributing x cases of paint per week. She incurs the following expenses: (1) Fixed costs of $1200 (2) An expense of $60x per week representing the cost of x cases to the dealer ($60 per case) (3) A cost of $x^2/24 per week for...
  14. N

    Question from Boyd's Optimization Book

    Hi, I am reading Convex Optimization from Stephen Boyd's book on my own and I am stuck at math he mentions on Pg. 157 of his book which can be found here. How does he write the following: sup{uTP^{T}_{i}x | ||u||2 ≤ 1} = ||P^{T}_{i}x||2 Thanks guys
  15. S

    Easy way to solve optimization problems

    Hi, So i don't need help on any specific problem, I was just wondering if there was an easy way to solve optimization problems in calc. I have no problem doing most of it, its just that coming up with the functions is my biggest problem. Can anyone give me advice on coming up with the problems.
  16. D

    Optimizing Pasture Fencing: Min. Fencing Length & Area

    Homework Statement A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough gas for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river? Should...
  17. MarkFL

    MHB Mark22's optimization question from another site

    mark22 wrote: Note: The OP had shown work, and correctly obtained the distance, but was having trouble finding the corresponding time(s). My response: I think I would approach this parametrically. Let 12:00:00 (am or pm) be time $\displaystyle t=0$ in minutes and $\displaystyle m(t)$...
  18. N

    Iteration, recursion, optimization,

    Actually, it is really as easy as it looks. Paradoxically, it may be helpful to regard it as a special case of: 1 2 3 ...n 1 1 2 3 ...n 1 2 3 ...n 1 1 2 3 ...n 1 2 3 ...n 1 2 3 ...n 1 1 2 3 ...n 1 2 3 ...n 1 2 3 ...n 1 2 3 ...n 1 1 2 3 ...n 1 2 3 ...n 1 2 3 ...n 1 2 3 ...n 1 2 3 ...n 1...
  19. F

    Optimization problem with a round lake

    1. A person from point A wants to get to point C diammetrically across a round lake. This person is on the shore and can walk at a rate of 4 mi/hr and row at a rate of 2 mi/hr. Which method should she use? 2. radius = 2 mi, triangle with angle θ has the points ABC 3. I started out...
  20. R

    How do I set up an optimization problem?

    How do I set up an optimization problem?? Homework Statement A closed reactangular box, with a square base x by x cm and height h cm. The surface area is 8cm^2. Find the maximum * I know how to find the critical points and everything else, but I don't know how set the problem up. I have a...
  21. A

    [Python] Optimization: determining gradient with variable window size

    Hi all, I'm not quite sure if this is the right place to post my question, so forgive me if its not... I've written a program in Python that analyses data that I got from a compression experiment (mechanical testing of rocks and such), and I've written a piece of code that estimates the...
  22. O

    Bivariate Function Optimization

    Homework Statement In general, I've been given a few functions of two variables, x and y. I have been asked to find all critical points by setting the gradient of the function equal to 0. Further we are asked to classify these critical points using some given rules regarding the Hessian...
  23. M

    Find the Best Optimization Method for Creating Test Groups - Mike

    I am looking for a term to describe a sort of "optimization" that I am trying to do. Hi everybody, I am a college student who unfortunately is not smart enough to be a math major. But nonetheless I am obsessed with statistics and numbers. I have a question: I need help figuring out a term...
  24. T

    MHB Maximizing x1+1.2*x2+1.5*x3 subject to Constraints | Linear Equations

    Hello guys I have to maximise x1+1.2*x2+1.5*x3 subject 2*x3<=60 2*x2<=45 2*x1+x2+3*x3<=80 x1>=20 x2+x3>=10 x1,x2,x3>=0 I am told that one of the constraints is redundant i.e one of the equations can be removed and then use the Simplex method to obtain the values for x1,x2,x3. The problem is...
  25. D

    Optimization problem with constraint

    PROBLEM STATEMENT: Determine if f(x,y) = x^2+y^2 has a maximum and a minimum when we have the constraint 2x^3+3x^{2}y+3xy^{2}+2y^3=1. (1) ATTEMPT TO SOLUTION: A standard way of solving these kinds of problems is by using the Lagrangian multiplier-method. It consists of comparing the gradient of...
  26. B

    Mathematica Mathematica - Optimization Over an Elipse

    Hello, I need some assistance with Mathematica. I'm very new to the software, and am not very familiar with the programming language. I guess I'm a little bit lost as to where to go after defining my function. I'm not sure how to apply constraints, or really how to jump in and tackle this...
  27. P

    Optimization and Related rates

    Homework Statement A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks 20 miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is...
  28. R

    Truss optimization and analysis

    We were given a small project to do before a test in my engineering statics class, i understand the goals of the project and understand the steps i need to take (i believe) however i don't understand how to simplify the distributed load into something i can anylize as we haven't seen anything...
  29. countzander

    Maximizing z=xy^2-5 on a Bounded Region in the xy-Plane

    Homework Statement Find the global max/min for z=xy^2 - 5 on the region bounded by y=x and y=1-x^2 in the xy-plane. Homework Equations The Attempt at a Solution I found the critical point of z=xy^2 - 5 at (0,0), but I do not know how to relate this to the boundary.
  30. D

    Kuhn-Tucker Optimization Problem

    Well, I've been working on this problem, but I can't get the right path to the solution. Homework Statement "Consider the following version of the Ehrlich economic model of crime where an individual has the expected utility function: U = p ln(Iu) + (1 - p) ln (Is). p = objective...
  31. A

    Path optimization for a race course

    Ok, so here is my question In grade school we all learned that the shortest distance between two points is a straight line. We also know from intuition that cars can't withstand infinite amounts of radial or tangential acceleration otherwise the car will skid i.e traction circle. Via this...
  32. S

    Optimization & singular Hessian matrix

    I am trying to figure out how the least squares formula is derived. With the error function as Ei = yi - Ʃj xij aj the sum of the errors is SSE = Ʃi Ei2 so the 1st partial derivative of SSE with respect to aj is ∂SSE / ∂aj = Ʃi 2 Ei ( ∂Ei / ∂aj ) with the 1st partial derivative of...
  33. A

    How does the concept of optimization in physics relate to real-world scenarios?

    Sometimes I wonder why the optimization conditions you can find mathematically, show how a system behaves in nature. As an example you can calculate mathematically what curve a hanging rope will form such that its potential energy is minimized. But how do you know that there does not exist...
  34. L

    Partition techniques optimization

    Hello guys, I work on a final project study discussed the use of optimization methods. The project in question consists in partitioning a grid dimensions NXM grids in dimensions 3 X 3 (which share no box) to the extent possible, otherwise find grids of dimensions 3 x 3 which are dependent...
  35. A

    Help in Optimization using Genetic Algorithms

    Currently I am doing models for my system. I have finished all of them designing three defferent controllers (Feedback Linearization Controller FLC , Sliding Mode Controller SMC and Fuzzy Sliding Mode Controller FSMC).Now I am asked to optimize the FSMC using Gentic Algorithms but I have no idea...
  36. P

    Optimization Problem Maximum Volume

    Homework Statement a cylinder can be inscribed upright in a circular cone with radius 4 and height 7. What is the maximum volume of a cylinder that can be inscribed inside of the cone.Homework Equations Image of the problem http://s17.postimage.org/67akeibxb/Cylinder.png The Attempt at a...
  37. O

    Solving an Optimization Problem with Two Functions

    I have an optimization problem and I am looking for a method rather than a solution here. I'll state it in a general form. Let there be two functions: f_1(x_1,\cdots, x_n,y_1,\cdots, y_n ) and f_2(x_1,\cdots x_n,y_1,\cdots, y_n ). Maximize f_1 with regards to variables x_1,\cdots, x_n with...
  38. P

    Optimize Perimeter of Window: 3m² Rectangle & Triangle

    Homework Statement If the area of the window is 3m² what are the dimensions of a rectangle and (isosceles)triangle that will minimize the perimeter. http://s12.postimage.org/r1czis8jh/optimize.png --DIAGRAM Homework Equations Let AreaTotal=3 Let Areatotal=Atriangle+Arectangle Arectangle=lxw...
  39. G

    Fortran Programming trouble with fortran optimization code

    Hi everyone, I have some trouble adapting an existing fortran code for my application. In the following Module I am optimizing a value for two planes ( i = iplan one&two). In order to do that, I need to assign ten certain values from an external inp. file for each plane. I have two loops...
  40. L

    Closed-form solution of a quadratic optimization problem

    Hello, My question is as follows. Is it possible to obtain a closed form solution to \displaystyle \max_{\xi\ge 0, \lambda\ge 0}\,\, -\frac{1}{2}\||\xi\||^2 +(\xi,\,\lambda) -\frac{1}{2}\||\lambda\||^2 Here \xi and \lambda are vectors. Thank you.
  41. S

    Optimization of Road Repair and Construction Project

    Homework Statement Albany is 12km north of Rochedale. Bells Creek is 5km west of Albany. The road is to be repaired and repairs cost $96 000 per km. The cost of laying a new road is $120 000 per km. It has been decided to repair the old road from Rochedale as far as point K, and then to...
  42. S

    Optimization box with maximum volume?

    Homework Statement A manufacturer wants to design a box having a square base and a surface area of 108m^2. What dimension will produce a box with maximum volume?Homework Equations a=x^2+4xy V=x^2y a=108m^2 The Attempt at a Solution
  43. F

    Optimization of y=x^2 towards (2,1)

    Hi! Probably quite easy for you guys (Im not even sure I am in the right thread). The assignment is in constrained optimization, and we're supposed to use lagrange to find the point on the parabola y=x^2 which is closes to (2, 1). I've been trying for a while and can't seem to find the right...
  44. T

    COMSOL: Setting up simple optimization problem

    Hi all, i have some doubts about setting the COMSOL optimization module. My aim is quite simple, i should find the optimal value of a variable to minimize the objective equation. In particular i should set up the parameter C in this equation(which represents the outlet pressure of a laminar...
  45. T

    COMSOL: Problem about optimization applied to a laminar flow

    Hi all,i'm new here and i really hope that you could help me with this problem that is blocking the progression of my graduation thesis. I'm working on a 3D stationary simulation of brachiocefalic artery; I'm using laminar flow physics with a known velocity in inlet and a normal stress in...
  46. S

    Proving recursion relations. BFGS non linear optimization

    Homework Statement Please see attached thumbnail Here's what I know. 1)Bk is the Hessian 2) sk = \alpha*p 3)pk is the search direction 4) Alpha is the step size Homework Equations yk = \nablaf(xk+1) -\nablaf(xk Bk+1(xk+1-xk) = \nablaf(xk+1) -\nablaf(xk The Attempt at a Solution...
  47. V

    Optimization problem solved two ways (algebra or calculus)

    Homework Statement A life guard sitting on a beach at point A needs to get to point B (Hasselhoff fell out his inflatable rocking chair) as soon as possible. The lifeguard (Pamela Anderson) can run (on the shore in slow-motion, like in Baywatch) at a rate of 3 m/s and can swim at a rate of 1.5...
  48. M

    Optimization, multivariable calculus

    Homework Statement the base of an aquarium with given volume V is made of slate and the sides are made of glass. if slate costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the materials. Homework Equations The Attempt...
  49. M

    Optimization: Maximizing Profit

    Homework Statement Through market research, a computer manufacturer found that x thousand units of its new laptop will sell at a price of 2000 - 5x dollars per unit. The cost, C, in dollars of producing this many units is C(x) = 15 000 000 +1 800 000x + 75x^2. Determine the level of sales that...
  50. H

    What Is the Optimal Location for a Junction Box to Minimize Wiring Distance?

    Homework Statement Two isolated farms are 12km apart on a straight country road that runs parallel to the main highway 20km away. The power company decides to run a wire from the highway to the junction box, and from there, wired of equal length to two houses. Where should the junction box...
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