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. a255c

    Lagrange optimization: cylinder and plane intersects,

    Homework Statement The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin. Homework Equations $f(x) = x^2 + y^2 + z^2$ $h(x) = x^2 + y^2 = 1$ $g(x) = x + z = 1$ The Attempt at a Solution $\langle 2x, 2y, 2z \rangle...
  2. I

    Explain why this is correct (Optimization Problem)

    Homework Statement A piece of wire, 100 cm long, needs to be bent to form a rectangle. Determine the dimensions of a rectangle with the maximum area. Homework Equations P = 2(l+w) A = lw The Attempt at a Solution This is what I don't understand, the solutions that I saw from looking around...
  3. B

    List of quantitative methods for optimization

    Max: 3x + 5y s.t. x + 2y ≤ 5 x ≤ 3 y ≤ 2 x,y ≥0 By the simplex method, the profit is $14. Using sensitivity analysis I changed the RHS of the 1st constraint and keeping everything else constant, I get the best profit value of $19 at RHS of 7. What other methods can I use such as the...
  4. karush

    MHB Revisit t the ladder optimization problem

    This is a common homework problem but.. A fence $6$ ft high runs parallel to the wall of a house of a distance of $8$ ft Find the length of the shortest ladder that extends from the ground, over the fence, to the house of $20$ ft high and the horizontal ground extends $25$ ft from the fence...
  5. kevin2016

    A What is the closed-form solution using ALS algorithm to optimize

    C \in \mathbb{R}^{m \times n}, X \in \mathbb{R}^{m \times n}, W \in \mathbb{R}^{m \times k}, H \in \mathbb{R}^{n \times k}, S \in \mathbb{R}^{m \times m}, P \in \mathbb{R}^{n \times n} ##{S}## and ##{P}## are similarity matrices (symmetric). ##\lambda##, ##\alpha## and ##\beta## are...
  6. Mr. Rho

    Mathematica Rotation of 3D Plot using Euler angles

    So, I'm trying to plot a 3D "dipole" (an arrow with a small torus around it basically) in mathematica, and want to rotate it according to Euler angles... I use this code for the rotation matrix: rot[a, b, g] := RotationMatrix[g, {1, 0, 0}].RotationMatrix[b, {0, 1, 0}].RotationMatrix[a, {0, 0...
  7. Ornella

    Converting kW/h to kWh for Optimizing Fuel Cell Ramp-Up

    Hi everyone, I am working on a mathematical optimization model for a fuel cell. Currently I am facing a problem with the ramp-up of the cell. I have a modulation ramp of 4% of the nominal power (58.3 kW) per minute. My constraint in the model has to be in kWh (I have to precise that my...
  8. Devraj Dutt

    ANSYS Maxwell Optimetrics Error Handling

    A common problem with optimization of machines is that occasionally, some of the variable combinations will not be geometrically feasible. For example, if magnet angle is being parameterized in an IPMSM, it might be that at some point, the magnet skews so much that it juts out of the rotor...
  9. Bdhillon1994

    The End of the Ski Jump - Optimizing Launch Angle

    Homework Statement A ski jumper leaves the ski track moving in the horizontal direction with a speed of 25.0 m/s as shown in Figure 4.14. The landing incline below her falls off with a slope of 35.0°. Where does she land on the incline? I've attached an image of the problem, my work is below...
  10. vabm

    Enunciation/notation in utility maximisation model

    Hi Everyone. I am working on a model that I think can be defined as a utility optimisation problem but I'm struggling with the enunciation and notation. The model should describe how the utilities of a set of agents A={1,2,...,n} increase with the availability of a larger set of product types...
  11. ZenSerpent

    Single Phase Transformer Losses -- Hysteresis, Eddy Current Constants

    Hi. My colleagues and I are doing a research on transformers (single-phase) and we stumbled across the following equations involving hysteresis and eddy current losses: Wh = ηBmaxxfV where Wh = hysteresis losses η = Steinmetz hysteresis constant Bmax = maximum flux density x = constant...
  12. noowutah

    Specify function given certain constraints

    Let F:V\rightarrow{}\mathbb{R}^{+}_{0} be a differentiable function. V is the set of all positive real-valued 2\times{}2 matrices, so V=\left\{\left[ \begin{array}{cc} a & b \\ c & d \\ \end{array}\right]\mbox{ with }a,b,c,d\in\mathbb{R}^{+}\right\} Here are the two constraints for F...
  13. Y

    Finding the maximum value of a function

    Homework Statement Find the extremizing (maximum) value of the function f(x) = sin x / x using Newton's 1D method. Homework Equations [/B]The Attempt at a Solution I know the maximum point in this equation is (0, 1). When I differentiated the equation twice and used the formula above, I...
  14. perplexabot

    Changing to SDP (optimization)

    Hey all. Let me get right to it! I have the following objective function: \mathbf{minimize} \ \ trace((G^TG)^{-1}) I am trying to minimize it with CVX. I used schur complement to do the following: \begin{equation*} \begin{aligned} & \underset{G}{\text{minimize}} & &...
  15. G

    Optimization algorithm to apply to my system?

    I am at the moment working on a project in which I try to minimize the annual running costs of a chemical manufacturing plant. To predict annual running costs I created a model with over 50 inputs, including things such as the type of chemicals and equipment used at different points in the...
  16. patrickbotros

    A Find the minimum without Calculus or Graphing

    ƒ(ß)=.5sec(ß) + √[1+(sec2(ß)/4)+tan(ß)/√(2)] Without graphing it or using calculus find the minimum. I already know the answer but want to know how to do it. It s at π/12 and is something like 1.5. First off this is NOT a homework problem. I already know the answer is something like 1.5 at π/12...
  17. T

    MHB Solve Optimization Problem: Minimize Cost of New Highway

    Now I hate optimization problems and I cannot figure this one out at all. 1) A city wants to build a new section of highway to link an existing bridge with an existing highway interchange, which lies 8 miles to the east and 10 miles to the south of the bridge. The first 4 miles south of...
  18. H

    Maximizing Constrained Optimization Problem

    Homework Statement There is a typo in the problem, ”R > Σ n i=1 σi − n max 1≤i≤n σi” which should be R > n max (1≤i≤n) σi − (Σ n i=1 σi ) Homework EquationsThe Attempt at a Solution Not sure where to go with part B or where to start...
  19. S

    Minimizing surface area of a shaker

    Hi, I have an mathematics assignment to do, and I wonder if the topic I have chosen is doable for me. I want to minimize the surface area of a cobbler cocktail shaker, and until now my plan was to get the curve equation for the side of it, and get the area equation from surface of revolution...
  20. T

    Optimization of windmill power with use of gears

    < Mentor Note -- thread moved to HH from the technical Engineering forums, so no HH Template is shown > Okay so I'm a freshman BE student and one of our first projects is designing a windmill that can produce a voltage of 5 for 2 seconds or longer. We are having trouble find the optimal gear...
  21. Greg Bernhardt

    How Does MIT's New Algorithm Revolutionize Optimization Speeds?

    http://news.mit.edu/2015/faster-optimization-algorithm-1023 PDF: http://arxiv.org/pdf/1508.04874v1.pdf
  22. TyroneTheDino

    Optimization of ellipse surrounding a circle

    Homework Statement Consider the ellipse ##\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1## that encloses the circle ##x^{2}+y^{2}=2x##. Find the values of a and b that minimize the area of the ellipse. Homework Equations ##Area=ab\pi## The Attempt at a Solution I begin by completing the square...
  23. D

    UCM to Projectile: Optimum Launch Angle/Velocity Dependence?

    I ran a simulation on WinPlot (see attached video) on my computer and was a bit surprised to see that the optimal launch angle of a projectile (with NO air drag) leaving uniform circular motion is dependent on the initial tangential velocity (or at least Winplot thinks it does). Can someone...
  24. L

    Optimization growth rate Question

    First post on these forums, thanks all for your help! Homework Statement It is estimated that the growth rate of the fin whale population (per year) is rx(1 - x/K), where r = 0.08 is the intrinsic growth rate, K = 400,000 is the maximum sustainable population, and x is the current population...
  25. T

    What software for optimization surface of 4 parameters?

    Hello, I have data for optimization that contains different values of 4 different parameters, and the respective function value based on them. I want to find the maximum of this function, while making sure it's a "stable" maximum, i.e., for nearby parameters, the value of the function shouldn't...
  26. C

    Can Linear Programming Solve Problems with Division Constraints?

    Hello, Below is a description written in Latex. I am not sure how to solve this problem. I am new to linear programming and, in fact, I do not know if it can be solved by linear constraints. Please guide. Thanks
  27. Z

    MHB [Calculus] Finished optimization problems. Would someone please check them?

    Hello, I've just finished two optimization problems for my calculus class. I would really appreciate it if someone could check my work for me. https://docs.google.com/document/d/1vcMKOg0oD0sKIrY_AzUpRrAbTsBl3nxq8vod_1dVqyE/edit?usp=sharing
  28. Y

    {Calculus 1} Optimization Problem

    Homework Statement A wire is divided into two parts. One part is shaped into a square, and the other part is shaped into a circle. Let r be the ratio of the circumference of the circle to the perimeter of the square when the sum of the areas of the square and circle is minimized. Find r...
  29. I

    Various ways of doing work to minimize entropy increase

    [Moderator's note: Recategorized thread to "Basic".] While driving alone through the beautiful scenery of Banff and Yoho national parks, a question formed in my mind. Which of these modes of slowing down a vehicle by an equal amount is likely to minimize the resulting overall increase in...
  30. P

    Algorithm for creating unique groups of elements

    Homework Statement so for a side task I'm supposed to assign people to groups for an icebreaker in python, can anyone give me links to theories that I could read up on or give me suggestion X number of people at my company signed up for a dinner roulette as a way to meet new people. Everyone...
  31. J

    Find Function/Transform for signal that minimizes CV of data

    Warning...this requires scripting and iteration, and is not theoretical -- it is a real problem I haven't been able to solve, but I'm sure someone here can... :-) Data: each .csv file is a test recorded at a time interval of 7.5Hz and each file has 3 columns. The first column is time in...
  32. Z

    MHB Calculus: Find Closest Point to (3,0) on y=x^2

    Hello, I've been assigned two calculus problems and have completed both of them. I'm pretty sure the first one is correct but I'm iffy on the second one. I would really appreciate it someone here could check my work on the second problem, and maybe even on the first problem if they have the...
  33. maistral

    Nonlinear constrained optimization - how?

    Perhaps the title says it all, but I should expand it more, I guess. So I am trying to explore more about constrained optimization. I noticed that there are very little to no formal (with examples) discussions on algorithms on nonlinear constrained optimization in the internet. They would...
  34. S

    A Function Optimization Problem

    Can anyone tell me straightforward information about a way to maximize a certain functional I[f]=\displaystyle\int_{X} L(f,x)dx such that the integral is bounded, T≥\displaystyle\int_{X}f(x)h(x)dx. I really know a minimal amount about functional analysis and calculus of variations, but I've...
  35. anemone

    MHB What is the Minimum Value of a Mathematical Function with Specific Constraints?

    Find the minimum of $\dfrac{1}{a-b}+\dfrac{1}{b-c}+\dfrac{1}{a-c}$ for real $a>b>c$ given $(a-b)(b-c)(a-c)=17$.
  36. Y

    MHB Stationary points in local optimization

    Hello again, I have a small problem. I am looking for local minimum and maximum points of the function: \[f(x,y)=3x^{2}y+y^{3}-3x^{2}-3y^{2}+2\] The first question was how many stationary points are there. I have found the derivatives by x and y: \[f_{x}=6xy-6x\] \[f_{y}=3x^{2}+3y^{2}-6y\]...
  37. W

    Can Poker Chip Arrangements Be Optimized for Maximum Profit?

    Hey everyone. I was pondering how best to optimize a chip arrangement for a poker game. This is the scenario I've thought up: There are 4 denominations of colored chips with a set value. White (W) = 0.05 Red (R) = 0.25 Blue (B) = 1.00 Green (G) = 5.00 A player wants to purchase 40 dollars...
  38. Y

    MHB Solve Optimization Problem: Find Min & Max of f(x,y)

    Hello all I am trying to find minimum and maximum of the following function: \[f(x,y)=4x^{2}-y^{2}-xy-2x+6y\] under the constraints: \[y=4-2x\] \[x\geq 0\] \[y\geq -2\]I tried solving this problem using the method of the method of bounded and closed domain, understanding that the...
  39. N

    Graphical meaning of tangent in optimization problem

    In a trivial optimization problem, when seeking the value of x2 that minimizes y(x2)/(x2-x1), the solution is graphically given by the tangent line shown in the figure. I'm having a lot of difficulty understanding why this is true, i.e., the logical steps behind the equivalence supporting the...
  40. S

    Optimization methods with bivariate functions

    Hi, I have the following equation: f(z)=g(z)+b*u(z) where z=(x,y) i.e. bivariate,b is a parameter, u(z) the uniform distribution and g(z) a function that represents distance. By considering for a momment b=0, min(f(z)) can give me the location of the minimum distance. However because I want...
  41. S

    Grouping constrained optimization

    Hi all, I am looking for an efficient solution to solve the following problem. Can anybody help? Assume a set S of elements ki and a set V of possible groupings Gj. A grouping Gj is a subset of S. Associate a weight wij to each mapping ki to Gj. The weights are infinite if ki ⊄ Gj, and finite...
  42. Emmanuel_Euler

    Solve Optimization Problems: Check Solutions

    How can i check my solution in optimization Problems.??
  43. P

    Optimization using Newton's method gradient hessian

    Hello, This is my first post here. So I hope I'm posting in the right place, sorry if not. http://homes.soic.indiana.edu/classes/spring2012/csci/b553-hauserk/Newtons_method.pdf I am trying to solve the following numerical optimization function using Netwon's Method: So, if I have the gradient...
  44. M

    Engineering Books on optimization problems

    I am in search of (elementary) books which contains subject on the optimization problem. I'm not an electrical engineering student though so I don't know if that subject is typically taught in undergrad or grad level. But in case there is undergrad book that meets that condition, I would prefer...
  45. TheAstroMan

    Crossing a river with a current (optimization)

    1. A man is on one side of a river that is 50 m wide. He is trying to get to someone directly on the other side. There is a current flowing down the stream at 2.4 m/s. His swimming speed is 3 m/s and his walking speed is 10 m/s . What is the best angle for him to swim at to have the fastest...
  46. K

    Optimization Problem: Minimizing rectangle dimensions

    Hi guys, I'm a high school senior currently in calculus and vectors. We're in our application unit right now, and I'm having quite a bit of trouble with problems that give the desired volume/area, and then ask you for the minimum dimensions required for said volume. One notable problem that I am...
  47. C

    Optimizing Isosceles Triangle Problem: Find Min. Sum of Distances

    Homework Statement An isosceles triangle has a base of length 4 and two sides of length 2sqrt(2). Let P be a point on the perpendicular bisector of the base. Find the location P that minimizes the sum of the distances between P and the three vertices. Homework Equations N/A The Attempt at a...
  48. kostoglotov

    Discontinuity of a constraint in Lagrange Method

    Homework Statement My question is quite specific, but I will include the entire question as laid out in the text Consider the problem of minimizing the function f(x,y) = x on the curve y^2 + x^4 -x^3 = 0 (a piriform). (a) Try using Lagrange Multipliers to solve the problem (b) Show that the...
  49. R.C

    Power Generation (Optimization) in a Car Engine

    **Long Post** 1. Homework Statement The Mazda MX5 (2001-2005) comes with two engine sizes, 1.6 litre and 1.8 litre which are rated as 110 bhp and 146 bhp respectively at 6,500 rpm. You are required to choose one of the two engine sizes and analyse the system with a view to improving the power...
  50. V

    Material Studio (geometry optimization)

    Hi. I am using Material Studio (MS) version 4.3 (yeah I know it's old but that's all I've got) and I have been trying to calculate this energy of a specific compound (N,N-dimethylformamide). I've learned that there are 3 ways on how to calculate energy using MS; Forcite, Dmol3 and Minimizer...
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