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. G

    Optimization - rectangle inscribed in a right triangle

    Homework Statement A rectangle is to be inscribed in a right triangle having sides 3 cm, 4 cm and 5 cm, as shown on the diagram. Find the dimensions of the rectangle with greatest possible area. Homework Equations 1. x^{2}+y^{2}=w^{2} in terms of w=\sqrt{x^{2}+y^{2}} 2...
  2. B

    Optimization problem that makes no sense

    Homework Statement The cost of fuel to propel a boat through the water (in dollars per hour) is proportional to the cube of the speed. A certain ferry boat uses $100 worth of fuel per hour when cruising at 10 miles per hour. Apart from fuel, the cost of running this ferry (labor...
  3. T

    MHB Finding Max & Min of Optimization Problem: How To Distinguish

    How are Finding the max and min of a optimization problem different. and how do you distinguish them in an equation?
  4. Q

    Find the Speed of a Car Approaching at 190mph: Optimization Problem

    A police is flying at 150 mph at a constant altitude of .5 miles above a straight rode. The pilot uses radar to determine that an oncoming car is at a distance of exactly one mile from the helicopter, and that this distance is decreasing at 190 mph. Determine the speed of the car. This is a...
  5. D

    MHB What is the Optimal Rectangle in a Semicircle with Radius R?

    I'm really stuck on this problem. Could anyone provide some help? Find the length and width of the rectangle of largest area that can be inscribed in a semicircle of radius R, assuming that one side of the rectangle lies on the diameter of the semicircle. Also, find the area of this rectangle...
  6. Vanrichten

    MHB Multiple Integral and Optimization in 3D

    In this problem I have drawn out the region specified and noticed two sets of parallel lines indicating to me that a change of variable(u and v) are able to be used to solve this integral. I decided that u=y-x and v = -2x-y then solving for x and y I obtain x= (u-v)/3 and y = (4u-v)/3 From...
  7. I

    Calculating Distance Between Point and Line in 3-Dimensional Space

    I am completely self taught in Calculus, but I would like to have a better understanding of it before I go to College. I was looking up problems and came across this: What is the distance d between the point A(2,-6,1) and the line l through B(3,4,-2) and C(7,-1,5)? I attempted this by finding 3...
  8. M

    Chemical Reactor Design Optimization

    Homework Statement Homework Equations The Attempt at a Solution For this problem, my group is having difficulty figuring out how much Styrene is reacting, since we don't know the molecular weight of polystyrene, we can't find the moles of polystyrene, which would be used to aid in finding the...
  9. R

    Optimization problem (imaging science)

    Homework Statement Assume a transmission imaging system with the following parameters: focal spot PSF equal to a Gaussian with FWHM = 0.5 mm, a detector PSF equal to a Gaussian with FWHM = 0.1 mm, source-to-detector distance = 100 cm. Compute: 1. the optimal source-to-object distance 2...
  10. Q

    Optimization of Area; Inscribed Rectangles

    Homework Statement http://i.minus.com/jbkHm5oH1LfQ1k.png Homework Equations The area of a rectangle is its base times its height. The Attempt at a Solution The rectangle is inscribed. Its area is 2xy. I can substitute in the equation of the semicircle to get rid of the y-term...
  11. Q

    Optimization - Why is there a constant term in the time equation?

    Optimization - Why is there a constant term in the time equation?? Mod note: The OP apparently answered his own question. Homework Statement 1) A man can run at 8 km/hr and swim at 4 km/hr. He is currently 6 km from the shore of a lake, which is due east from him. He wants to get to a point...
  12. anemone

    MHB Optimization With Two Constraints

    Hi MHB, I've come across this problem recently: If a²+b²=2 and (c-3)²+(d-4)²=1, find the maximum value of ad-bc. I haven't solved it and actually, I don't know of any algebraic method I could use to solve it and I am wondering if this problem is doable with the Lagrange Multiplier...
  13. C

    Parameter Scaling for Optimization

    So I am still confused about how to applying scaling of parameters to a general optimization problem. Let's say I am trying to do maximum likelihood estimation. I understand how to find the scaling matrix (assuming we restrict it to diagonal form) and that the Hessian should be close to the...
  14. S

    Looking for ray tracing and surface optimization suggestions

    I am a physicist who is interested in acoustic guitar design (additionally I make custom made guitars and violins). The problem I have is in acoustic guitar design, for simplicity imagine a box 20"X15"X3" (length, width, depth). On one of the 20X15 surfaces (the top) cut a 3" hold 1/3 from...
  15. H

    Can I Optimize X for a Given Level Curve with Partial Derivatives?

    I have a function z=f(x,y) that is reasonably well behaved (single global maximum). I can readily compute the value of z as well as partials of z with respect to x and y. I can also quite easily find the maximum. The challenge is to find the maximum and minimum values of x where c =...
  16. Y

    Network optimization / Min cost flow problem

    Homework Statement Consider a power grid consisting of electricity producers that are connected to consumption points on the grid. The consumption points are affiliated with regional retail power companies that then distribute the power to their end users. The undirected graph (attached)...
  17. R

    Interested in optimization for scientific computing, where to start?

    I've just started programming with Python this summer, and I'm taking a course in computational physics this semester. I've been really enjoying it and programming in general, but I don't have much knowledge in computer science save for intro stuff (string/list methods, functions/classes...
  18. Y

    Converting a linear optimization problem with absolute values

    Homework Statement Here is an alternative approach to handling absolute value terms as the decision variables: abs(x) is the smallest value z that satisfies x \leq z and -x \leq z. Using this,convert the following into a lp Min 2x1 + 3abs(x2) S.T x1 + x2 \geq 6 Homework Equations Here is a...
  19. MarkFL

    MHB BBandaRR's questions at Yahoo Answers regarding optimization

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  20. A

    Searching 4 graph optimization software

    I'd like 2 solve the following problem (well, routinely solve a bunch of such problems): Let us have a number of points (vertices), that can be interconnected. Not any 2 points are connected. Each connection is assigned a value. I want 2 find the maximum path in the graph, that is, the one with...
  21. Y

    Vapor-Compression Refrigeration System Optimization

    A simple vapor compression system. Based on constitutive relationships between pressure, temperature, enthalpy, etc., the system has four degrees of freedom. The two pressures in the system(p1 and p2), the level of superheat at the compressor outlet (T2) and the masss flow rate of the...
  22. MarkFL

    MHB Kendra Leota's Calculus Questions @ Yahoo Answers

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  23. H

    Probability from an optimization problem

    I have a cost function which consists of sum of a set of quadratic loss plus a term which regularize function. my problem is: is there any way to infer probability from such a cost functions?
  24. A

    What exactly is Engineering Optimization

    What exactly is "Engineering Optimization" I have the opportunity to take a grad class entitled "Engineering Design Optimization" this fall and I am curious about the subject. I have never taken an optimization class before and to be honest I don't quite understand why optimization is...
  25. R

    Refrigeration Heat Exchanger Area Ratio Optimization

    The equation for heat transfer is: 1. Q = U.A.(TL-T1) Where: Q= heat transferred (W). U= overal h.t. coefficient. A= area of heat exhanger (m^2). TL= Temp. of refrigerated space (Kelvin) T1= Low temp. of refrigerant= refrigerant in evaporator (K) I want to eventually find the maximum heat...
  26. MarkFL

    MHB Optimization Challenge - Poles and Wires

    Suppose you have two poles separated by the distance $w$, the first of height $h_1$ and the second of $h_2$, where $0<h_1<h_2$. You wish to attach two wires to the ground in between the poles, one to the top of each pole, such that the angle subtended by the two wires is a maximum. What portion...
  27. D

    MHB Maximizing Area: Solving an Inscribed Circle Problem in an Isosceles Triangle

    Hey Everyone, I'm having trouble setting up this word problem. I know that area of a circle is pir2 and area of a triangle is 1/2bh but for some reason I can't find a way to combine these equation together. What are the radius and area of the circle of maximum area that can be inscribed in an...
  28. MarkFL

    MHB ?'s questions at Yahoo Answers regarding optimization

    Here are the questions: I have posted a link to this topic so the OP can see my work.
  29. MarkFL

    MHB D4ncelife's question at Yahoo Answers regarding optimization of trapezoidal area

    Here is the question: I have posted a link there so the OP can see my work.
  30. J

    Optimization: maximum curved surface area of a cylinder in sphere

    θHomework Statement The attached diagram depicts a sphere with several variables: the height of the cylinder, the radius of the cylinder and an angle. All that has been given to me is the hypotenuse of a triangle used. Homework Equations To my knowledge I was told to use 2sinθcosθ=sin2θ...
  31. I

    MHB Function of symmetry character for comparing two optimization methods

    First refer to this question:Is this function "$h$" symmetric of the plane $x=y$? - Mathematics Stack ExchangeThe main problem is to find an appropriate (objective) function (a curve) to compare the implicit filtering method and Nelder-Mead method for optimization. The backgrounds of these...
  32. B

    Real life Optimization problem

    Hi context: The USD/CNH currency pair has been trending down at an almost linear rate past 3 years, it's safe to assume it will continue to trend downwards for the short term foreseeable future as the Chinese govt slowly allows the Yuan to appreciate to its real value. The problem: Every...
  33. A

    Which shape provides the largest area for a given perimeter?

    The problem is posted here: http://nrich.maths.org/5673 I tried subbing in values to get the biggest area: perimeter ratio possible. I used an arbitrary value, 12, for perimeter. For the triangle, I took out the hypotenuse and so A = 1/2(x)(y) where x+ y = 12. Its area is biggest when x = y = 6...
  34. L

    Help with an engineering optimization problem?

    Homework Statement The link to the problem is here: http://i.imgur.com/wyOsoSB.png The Attempt at a Solution I'm not completely sure about my work so far so please bear with me. My professor is very poor at professing the subject so I'm trying to learn from the book. Please let me know...
  35. B

    Optimization: find zeros of a derivative

    Homework Statement Find the maximum point of P(h)=-10h+4410-(6800/h) Homework Equations P(h)=-10h+4410-(6800/h) The Attempt at a Solution P(h)=-10h+4410-(6800/h) P'(h)=-10+(6800/h^2) P'(0)=-10h^2+6800 10h^2=6800 Divide both sides by 10: h^2=680 and sqrt both sides: h=26.1
  36. MarkFL

    MHB Zhina's question at Yahoo Answers regarding optimization subject to constraint

    Here is the question: Here is a link to the question: CALCULUS needHELP WITH THIS PLEASE! APPRECIAT EIT -? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  37. F

    Finding Arc Length in Optimization Problem

    Homework Statement Joe is traveling from point A across a circular lake to a cabin on the other side at point B. The straight line distance from A to B is 3 miles and is the diameter of the lake. He travels in a canoe on a straight line from A to C. She then takes the circular trail from C to...
  38. NATURE.M

    Maximizing the Sum of Squares for Two Non-Negative Numbers with a Given Sum

    Homework Statement Two non-negative numbers are chosen such that their sum is 30. Find the numbers if the sum of their squares is to be a maximum. Homework Equations The Attempt at a Solution let a,b represent the two non negative numbers a=x b=30-x So, x^2+(30-x)^2=s, where...
  39. W

    Optimization Problem, Laying down electrical lines (Multivariable)

    Homework Statement Homework Equations The Attempt at a Solution I am not really sure how to optimize it from here. What should I be taking the derivative of the price function with respect to ??
  40. F

    Optimization Problem involving a wire

    1. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum? 2. A_{s}(x) = x^{2}, A_{t}(x)=\frac{\sqrt{3}}{4}x^{2}...
  41. I

    Proof of convergence theory in optimization

    Homework Statement The question is: Suppose that lim x_k=x_*, where x_* is a local minimizer of the nonlinear function f. Assume that \triangledown^2 f(x_*) is symmetric positive definite. Prove that the sequence \left \{ f(x_k)-f(x_*) \right \} converges linearly if and only if \left...
  42. I

    MHB Proof of convergence theory in optimization

    The question is:Suppose that lim $x_k=x_*$, where $x_*$ is a local minimizer of the nonlinear function $f$. Assume that $\triangledown^2 f(x_*)$ is symmetric positive definite. Prove that the sequence $\left \{ f(x_k)-f(x_*) \right \}$ converges linearly if and only if $\left \{ ||x_k-x_*||...
  43. P

    Optimization problem using derivatives

    Homework Statement We want to make a conical drinking cup out of paper. It should hold exactly 100 cubic inches of water. Find the dimensions of a cup of this type that minimizes the surface area. Homework Equations SA = pi*r^2 + pi*r*l where l is the slant height of the cone. V =...
  44. MarkFL

    MHB Eggy's question at Yahoo Answers regarding optimization with constraint

    Here is the question: Here is a link to the question: Calculus Max Min problem help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  45. MarkFL

    MHB Chelsea's question at Yahoo Answers regarding functions and optimization

    Here is the question: Here is a link to the question: I need help on a HW problem - I got the first two parts right, but am having trouble finishing the last two? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  46. Petrus

    MHB Multivariable optimization with constraint

    Calculate biggest and lowest value to function f(x,y)=x^5y^4e^{-3x-3y} In the triangle has vertices in points \left(0,0 \right),\left(6,0 \right) and \left(0,6 \right) Before I start I want to warn that I used google translate in the text 'In the triangle has vertices in points' Progress: I...
  47. MarkFL

    MHB USCTrojanTennis' question at Yahoo Answers regarding optimization

    Here is the question: Here is a link to the question: Show that the minimum and maximum points for y=f(x) occur among the min and max points for y=f^2(x).? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  48. MarkFL

    MHB Danielle's question at Yahoo Answers regarding optimization

    Here is the question: Here is a link to the question: Helpppppppppppp please !? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  49. G

    Multi Variable Optimization Problem

    I have a problem that I normally find solutions to via trial and error, and they usually aren't optimized, but was wondering if there is a better way to solve this and optimize. My application is specific but this is the best way I can describe the problem. Forgive me if it doesn't make...
  50. MarkFL

    MHB Maria's question at Yahoo Answers regarding optimization

    Here is the question: Here is a link to the question: OPTIMIZATION PROBLEM? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
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