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.

View More On Wikipedia.org
Recent contents Top users
  1. P

    Optimization -writing and solving the equations

    Hi Im trying to write an equation for the question below. Could someone please point me in the right direction with writing it? Homework Statement An island is 4km from the nearest point p on the straight shoreline of a lake. if a person can row a boat at 3km/h and walk at 5km/h where...
  2. N

    LaTeX Optimization problems in LaTeX

    I am trying to write something like: minimize {w \in \mathbb{C}^N} You can see it in the attached file. It is written in MathType. I want to do the same in LaTeX. One way (not so correct) is to use "min" instead of "minimize": \displaystyle \min_{w \in \mathbb{C}^N}
  3. N

    Optimization Problem Homework: Need Help Finding Expression for h

    Homework Statement Here is the exact problem, in order to avoid confusion The Attempt at a Solution I know that I have to find an expression for "h" and substitute it back into the Volume, but the way I do it, it just becomes way too messy to find the derivative. Any ideas?
  4. R

    To which complexity class does this optimization problem belong?

    I'll try to be as abstract as possible, but where needed, I'll give some concrete examples. If you have any questions, please ask. Note, I'm doing this for my hobby, not for any sort of homework. I've only followed an introductory course on computational complexity, so I'll let that be my...
  5. S

    How can I optimize my homework solutions for efficiency and accuracy?

    Homework Statement http://img7.imageshack.us/img7/1826/43544187.jpg Homework Equations The Attempt at a Solution whats wrong with my answers? everything looks right to me... :S
  6. S

    Question on optimization and limits

    You are designing a rectangular poster to contain 50 cm2 of printing with margins of 4 cm each at the top and bottom and 2 cm at each side. What overall dimensions will minimize the amount of paper used? What i did was let the length and breath of the whole poster to be x and y so the area...
  7. A

    Optimization and related rates trig.

    Homework Statement A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at point C diametrically opposite A on the shore of the lake in the shortest time possible. She can walk at 4 mph and row a boat at 2 mph. To what point on the shore of the lake should...
  8. S

    Rectangles: Smallest Perimeter for Given Area, Greatest Area for Given Perimeter

    Homework Statement a) Show that of all the rectangles with a given area, the one with the smallest perimeter is a square. b) Show that of all the rectangles with a given perimeter, the one with the greatest are is a square. Homework Equations As=x2 AR=xy Ps = 4x PR= 2x+2y The...
  9. P

    Optimization largest possible volume problem

    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. Volume of a cylinder = (pi)(r^2)h Volume of a cone = (1/3)(pi)(r^2)h Homework Equations Volume of a cylinder = (pi)(r^2)h Volume...
  10. T

    Brine Tank Optimization Problem

    Homework Statement A tank is initially filled with 1000 litres of brine, containing 0.15 kg of salt per litre. Fresh brine containing 0.25 kg of salt per litre runs into the tank at the rate of 4 litres per second, and the mixture (kept uniform by stirring) runs out at the same rate. Show...
  11. M

    Solve Optimization Problem for Max Cylinder Volume w/ 36cm Perimeter

    Just want to make sure I am doing it correct! a Rectangle sheet of perimeter 36cm with dimensions x (vertical) and y (horizontal) is to be rolled into a cylinder where x= height and y= circumference. what values of x and y will give largest volume? Write volume in terms of only one variable...
  12. E

    Optimization problem - Trouble differentiating function

    Optimization problem -- Trouble differentiating function Homework Statement The efficiency of a screw, E, is given by E=\frac{(\Theta - \mu\Theta^{2})}{\mu + \Theta} , \Theta > 0 where \Theta is the angle of pitch of the thread and \mu is the coefficient of friction of the material, a...
  13. S

    Suggest some optimization problems for me, please.

    Hi, this is my first post and most certainly not my last. I'm a young Mechanical Engineering major and I love math and physics, but on with my topic... I'm in Calc I and we've been assigned an extra-credit group project where to do present either a related rate or an optimization problem...
  14. D

    Optimization of a folded piece of paper

    Homework Statement If you take an 8.5in by 11in piece of paper and fold one corner over so it just touches the opposite edge as seen in figure (http://wearpete.com/myprob.jpg ). Find the value of x that makes the area of the right triangle A a maximum? Homework Equations A = 1/2(xy)...
  15. P

    Functional optimization problem

    Homework Statement Maximize the functional \int_{-1}^1 x^3 g(x), where g is subject to the following conditions: \int^1_{-1} g(x)dx = \int^1_{-1} x g(x)dx = \int^1_{-1} x^2 g(x)dx = 0 and \int^1_{-1} |g(x)|^2 dx = 1. Homework Equations In the previous part of the problem, I computed...
  16. S

    Minimization - optimization alg. or equation alg.?

    Hello everybody! I guess my question is mainly concerned with numerical algorithms... Given a problem of the form min w = f(x) subject to g1(x)=0 : : gn(x)=0 where x is a m x 1 vector, n < m. From a numerical standpoint, how can I know whether it is preferably to solve it by setting up the...
  17. Tclack

    Optimization, cylinder in sphere

    Homework Statement Find the dimensions(r and h) of the right circular cylinder of greatest Surface Area that can be inscribed in a sphere of radius R. Homework Equations SA=2\pi r^2+2\pi rh r^2 + (\frac{h}{2})^2 = R^2 (from imagining it, I could also relate radius and height with r^2...
  18. F

    Optimization Math Help: Finding Minimum Enclosed Area with Wire Cut

    Homework Statement A piece of wire 8 cm long is cut into two pieces. One piece is bent to form a circle, and the other is bent to form a square. How should the wire be cut if the total enclosed area is to be small as possible? Keep \pi in your answer. Homework Equations A= \pi r^{2} A= lw...
  19. J

    Optimization of the area of a triangle

    Homework Statement What is the maximum area of an equilateral triangle and a square using only 20ft of wire? Homework Equations 20=4x+3y x=\frac{20-3y}{4} A=x^2+\frac{1}{2}y^2\sqrt{3} The Attempt at a Solution So then A=\frac{400-120y+9y^2}{16}+\frac{y^2\sqrt{3}}{4}...
  20. S

    Oil Refinery: Optimization Problem

    Homework Statement A drilling rig 12 miles off shore is to be connected by a pipe to a refinery onshore, 20 miles down the coast from the rig. If underwater pipe costs $40,000 per mile and land based pipe costs $30,000 per mile what value of x and y would give the least expensive connection...
  21. N

    Who wants to help me maximize the area of a kite? (Optimization)

    I just can't figure this problem out. Homework Statement You have four pieces of wood, two with length a and two with length b, and you arrange them in the shape of a kite (pieces of equal length placed adjacent to each other). You want to build a cross in the middle as a support. How...
  22. T

    Searching for software for logic optimization

    I've got quite an unusual hobby project and so far, after couple of nights googling, I haven't found software that would fit the bill. I've got the truth table representing what I'd like to do and can minimize & map it to gates using Logic Friday. The problem is, I don't have NOR or NAND...
  23. N

    Optimizing 2D Area with Objects - CS Graduate Student Seeking Advice

    Hello, I am a CS graduate student, and I have a curious optimization problem which i need to solve, and have no idea where I should be looking for techniques for solving it. I have searched much material on optimization techniques, but still am not sure which subject this falls under. I would...
  24. D

    Optimization with Constrained Function

    Homework Statement 1000m^2 garden. 3 sides made of wooden fence. 1 side made of vinyl(costs 5x as much as wood). Length cannot be more than 30% greater than the width. Find the dimensions for the minimum cost of the fence. Homework Equations 1000 = LW C = 2L + W + 5W The...
  25. S

    Linear algebra/ optimization proof

    Homework Statement A vector d is a direction of negative curvature for the function f at the point x if dT \nabla ^2f(x)d <0. Prove that such a direction exists if at least one of the eigenvalues of \nabla ^2 f(x) is negative The Attempt at a Solution Im having trouble with this...
  26. S

    How Can You Optimize Fence Length Without Using Derivatives?

    Hey guys this isn't exactly a homework question. I'm helping my girlfriend with her grade 12 college level math course. When i was in grade 12 i took calculus.. and she called me and asked for help with optimization. I don't think in her class they are learning about calculus so how would you...
  27. W

    Cylindrical optimization problem

    A closed cyliindrical container has a volume of 5000in^3. The top and the bottom of the container costs 2.50$in^2 and the rest of the container costs 4$in^2. How should you choose height and radius in order to minimize the cost? v=pi(r)^2 Unfortunately my attempt at this problem is...
  28. W

    Pretty simple optimization problem

    A closed box with a square base is to contain 252ft^3. The bottom costs 5$ per ft.^2, the top is 2$ft^2 and the sides cost 3$ft^2. Find the dimensions that will minimize the cost. As for equations we have v=lwh and I'm not sure as to how to find the next relative equation. I just...
  29. R

    Optimization expression Problem

    Homework Statement Hello, Can you help me to understand the question..Ineed clarification and hints to solve the question.. A circle and a square are to be constructed from a piece of a wire of length l. 1-give an expression for the total area of the square and circle formed...
  30. D

    Optimization - Find dimension of a cup that uses least amount of paper

    Homework Statement A cone-shaped paper drinking cup is to be made to hold 30 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. Homework Equations volume of a cone (1/3)(pi)(r^2)(h) = 30 SA of a cone pi(r)[sqrt(r^2 + h^2)] The Attempt at...
  31. P

    Optimizing Area with Perimeter Constraints

    Homework Statement the sum of the perimeters of an equilateral triangle and a square is 10. Find the dimensions of the triangle and the square that produce a minimum total area. Homework Equations The Attempt at a Solution my problem is finding the primary and secondary equations...
  32. P

    Optimization problem - minimize cost

    Optimization problem -- minimize cost Homework Statement A motor is floating on a buoy in a large pool with a perfectly flat shoreline. The motor is 3ft from the edge of the pool. A circuit to control operation of the motor is located 100ft from the closest point of the motor to the edge...
  33. B

    Optimization car travel problem

    I have been working on this problem for 30 min and can;t seem to get anywhere. Question You have a amphibian vehicle which can travel 20 mph on the water and 52 mph on the land. You must find the quickest point from point A to C. -A is 20 Miles east of point B and is all land. -C is...
  34. D

    Maximizing Cone Volume Inside a Sphere

    Homework Statement Find the dimensions of the right circular cone of maximum volume that can be inscribed in a sphere of radius 15cm.Homework Equations The Attempt at a Solution let r be radius of circular base of cone let y be height of small right triangle let h be height of cone r^2 +...
  35. D

    Cylinder Optimization for a Minimum?

    Question: Cost of producing cylindrical can determined by materials used for wall and end pieces. If end pieces are 3 times as expensive per cm2 as the wall, find dimensions (to nearest millimeter) to make a can at minimal cost with volume of 600 cm3. Relevant equations: a) V=600cm3=[pi]r2h b)...
  36. T

    Optimization using Lagrange multipliers

    1. Homework Statement [/b] f\left(x,y\right) = x^2 +y^2 g\left(x,y\right) = x^4+y^4 = 2 Find the maximum and minimum using Lagrange multiplier Homework Equations The Attempt at a Solution grad f = 2xi +2yj grad g= 4x^3i + 4y^3j grad f= λ grad g 2x=4x^3λ and 2y=...
  37. G

    [Materials studio]geometry optimization

    Dear friends: anyone who used the materials studio software before? these days, i used the software to do the geometry optimization of a compound's structure. it has already taken about 30hours to do the optimization job till now and it has not finished yet. i am really anxious and don't know...
  38. Q

    Optimizing a Linear Function with Constraint: A Tutorial on Lagrange Multipliers

    First let me clarify this is not a homework question. This part has cropped up as part of a small project i am doing on Cosmic microwave background. How would i go about minimizing the function f(x_{1},x_{2}...x_{n})=\Sigma*x_{i}*a_{i} subject to the constraint: \Sigma...
  39. D

    Optimization of a rectangular box with no top

    I am told in the problem that i am to minimize the amount of cardboard needed to make a rectangular box with no top have a volume of 256 in^3? I am to give dimensions of box and amount of cardboard needed. Can anyone help
  40. N

    Simple quadratic optimization problem

    Let P^ + ,P^ - ,I,Q \in R^{n\times n}, K\in R^{n\times 1}, M\in R^{1 \times n}, and assume that Q is positive definite, P^ - is positive semidefinite whence (MP^ - M^T + Q)^{ - 1} exists (where T denotes transpose). In what sense does K = P^ - M^T(MP^ - M^T + Q)^{ - 1} minimize the quadratic...
  41. T

    Optimization Problem with Cylinder

    Homework Statement The volume of a cylindrical tin can with a top and a bottom is to be 16\pi cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can? Homework Equations V=\pir2h The Attempt at a Solution So I...
  42. J

    Trade as an optimization problem

    Protectionism is often viewed as positive for the country that implements the protectionism but bad globally due to losses of efficiency and miss allocation of resources. Because protections measures are often met with counter protectionist measures countries try to trade freely and fairly for...
  43. P

    Optimization problem: Folding a triangle to minimize one side

    Homework Statement The upper right-hand corner of a piece of paper, 12 in by 8 in is folded over to the bottom edge. How would you fold it to minimize the length of the fold? In other words, how would you choose x to minimize y? Homework Equations None so far. The Attempt at a...
  44. E

    Optimization with maxima and minima

    A rectangle is to be inscribed in a right triangle having sides 6 inches, 8 inches, and 10 inches. Determine the dimensions of the rectangle with greatest area. I recently tried doing it and the answer was found by finding the slope and then using the first and second derivatives of the area...
  45. T

    How do you know if there isn't a solution to a calculus optimization problem?

    My teacher was saying that it is possible to have no solution to an optimization problem, and I was curious about how this could be possible. Could someone please explain and possibly give an example?
  46. C

    Optimization - Finding Minimum Between (0,0) and e^x

    Homework Statement Find the minimum distance from the origin to the curve y = e^x.Homework Equations Distance Formula The Attempt at a Solution http://carlodm.com/calc/prob6.jpg 5-6 bright Calculus kids in my high school grappled with this problem and we couldn't find an answer. Can...
  47. S

    Optimization in Several Variables

    F(x,y) = (2*y+1)*e^(x^2-y) Find critical point and prove there is only one. Use second derivative test to determine nature of crit. pt. I know the procedure in solving it: set partial derivatives to zero and solve resulting equations. And by second derivative test, if D>0, f(a,b) is local...
  48. C

    How to Calculate the Optimal Dimensions for a Rectangular Box from Cardboard?

    Homework Statement Question: You are planning to make an open rectangular box from a 10 by 18 cm piece of cardboard by cutting congruent squares from the corners and folding up the sides. 1) What are the dimensions of the box of largest volume you can make this way? 2) What is its...
  49. C

    Optimizing Fence Cost for Rectangular Field

    1. The problem statement A farmer wants to fence an area of 37.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? (Give the dimensions in increasing...
  50. C

    Optimizing Silver Plating Costs for a Square Box and Athletics Track

    Homework Statement 1. A closed box of square base and volume 36 cm^3 is to be constructed and silver plated on the outside. Silver plating for the top and the base costs 40 cnets per cm^2 and silver plating for the sides costs 30 cents per cm^2. Calculat ethe cost of plating the box so that...
Back
Top