What is Optimisation: Definition and 72 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
  1. Will26040

    MATLAB MATLAB Plug flow reactor optimisation Problem

    The assignment is to find the optimal operating temperature and maximum product concentration of reactant B, assuming a constant temperature across the PFR length. Please could someone help? thanks the reaction is a series reaction: A → B → C (liquid phase) Here is my current code which is...
  2. L Navarro H

    Proof that the exponential function is convex

    I try to proof it but i got stuck right here, i want your opinions Can I get a solution if i continue by this way? or Do I have to take another? and if it is so, what would yo do?
  3. Z

    MHB Maximizing Tr(A) & Unique Solution of Matrix A w/ Infinite Solutions

    Hello! I am new here, and I need (urgent) help regarding the following question: Let $\boldsymbol{A}_{(n\times n)}=[a_{ij}]$ be a square matrix such that the sum of each row is 1 and $a_{ij}\ge0$$(i=1,2,\dots,n~\text{and}~j=1,2,\dots,n)$ are unknown. Suppose that...
  4. Vigardo

    Optimisation software for trusses (or frames)

    Dear experts, What software is available (free or not) to optimise the member cross-sections of trusses or frames? Which one is your favorite? Would any of them optimise the topology and geometry? Ideally, the global buckling of the whole structure as well as the buckling and strength of...
  5. Shaw-krow

    Long distance broadcast, internet data packets across town?

    I got something i want to do for fun and help out a...person i know but i never finished high school so that's my degree level. i was wondering if there is a way under $100 to possibly use my internet or posibly 2 peoples internet connections on the other side of the town i live in, split up the...
  6. M

    B Optimizing Fair Distribution of Pooled Funds among Debtors

    Firstly, I'm not even sure how to frame this mathematically, but I'd be curious to know what kind of problem it is and what kind of subjects within maths it requires to be able to solve it. Here's my problem-- I've set it up in excel to help you visualise it. There is a different amount of...
  7. J

    I Choosing points from a set that produce the largest polygon

    I am trying to find the most efficient way to select points on a 2d plane from a set that maximizes the area of the of the shape they define when joined together. The points are all paired (sharing the same A->B vector), with these pairs also appearing mirrored about the origin. Here is an...
  8. Shakattack12

    What is the optimal number of items to produce for maximum profit in business?

    Homework Statement A manufacturer makes a batch of n items with the cost (in dollars) of each item being: n2-6n+35. The manufacturer sells the items for $50 each. How many items should be produced in each batch to maximise profit. Homework Equations Cost = C(n) = n2-6n+35 Revenue = R(n) = 50n...
  9. Shakattack12

    Maximizing Volume Formula for Given Box Dimensions and Cut Size

    Homework Statement A box is given with a L = 1.414w and W = w and a cut size = x. Find the general formula for the maximum volume. Homework Equations L = 1.414w - 2x W = w - 2x H = x The Attempt at a Solution V = x(1.414w - 2x)(w - 2x) V = 4x3 - 4.818x2w + 1.414w2 Apparently this isn't the...
  10. M

    A Summing simple histograms to recreate a more complex one

    I wouldn't be surprised if I've posted in the wrong section because in fact the reason for posting is to get help naming this problem. That being the first step to knowing where to look for a solution. Newbie to the forum so open to advice. The problem: I have a complex histogram and a...
  11. aphirst

    Failure of Optimisation for Nonlinear Equation Systems

    I wasn't sure into which category I should post this, so feel free to move it into a more appropriate place. As part of my work I'm solving a system of nonlinear equations, of a usual form: $$\vec{F}(\vec{X})=\begin{pmatrix}F_1(X_1, X_2, \cdots X_N) \\ F_2(\cdots) \\ \vdots \\...
  12. M

    I Combining Position and Velocity in PSO: How to Handle Different Units?

    Hello everyone,I have a short question about the PSO since I am a new comer to this field. how can we "add" position with velocity in the simple PSO algorithm, when they are of different units?
  13. Dominik Tugend

    I How to write main and sub objective for optimisation

    I have had this problem that I didn't know how to write it properly in this thread. Let's say I have two objective functions: minimize X minimize Y Let's say I want 1. as main objective and 2. a sub objective - by that I mean if there is a set of optimal solutions for 1., then final solution...
  14. P

    Calculus of variations question

    Okay, so I've run into a rather weird functional that I am trying to optimize using calculus of variations. It is a functional of three functions of a single variable, with a constraint, but I can't figure out how to set up the Euler-Lagrange equation. The functional in question is (sorry it's...
  15. W

    Find stationary points of a two variable function involving

    Homework Statement Find all stationary points of the function G(x, y) = (x^3)*e^(−x^2−y^2) Homework Equations fx=0 and fy=0 The Attempt at a Solution Gx = 3x^2*e^(-x^2-y^2) +x^3(-2x)e^(-x^2-y^2) = e^(-x^2-y^2)(3x^2-2x^4) Gx = 0 implies 3x^2-2x^4=0 x^2(3-2x^2)=0 hence x =0 ,+or-...
  16. F

    MHB Optimisation Problem (Global extreme points)

    Hi there everyone, wonder if anyone can help as I'm a bit confused. Ive been asked to find the global extreme point of f(x)=e^(x-1) - x. I have checked my answer against the solution and am correct and my working is as follows: f'(x) = e^(x-1) - 1 = 0. Therefore (x-1)=ln1 (which = 0) therefore x...
  17. P

    Why Do Nonlinear Functions Often Lead to Non-Convex Cost Functions?

    I am taking a course on linear regression online and it talks about the sum of square difference cost function and one of the points it makes is that the cost function is always convex i.e. it has only one optima. Now, reading a bit more it seems that non-linear functions tend to give rise to...
  18. 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...
  19. Khronos

    Optimisation - Critical Numbers for Complex Functions.

    Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows: Question: Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula: V = 999.87 −...
  20. J

    Need help on designing a conceptual HALE UAV

    Hi everyone, I am a High School student (final year) and am doing an extra-curricular project. Having complete autonomy on what we do the project on, I choose to design a High-Altitude Long-Endurance (HALE) UAV, such as the QinetiQ Zephyr [http://www.airforce-technology.com/projects/zephyr/] or...
  21. N

    MHB What are some recommended books for optimisation?

    Hi everyone, I will be taking a summer course on Optimisation - AMSI Summer School and was wondering if you could recommend any books. The course outline is: Week 1: Introduction to Optimization problems: classification and examples. Elements of convex analysis: convex sets and convex...
  22. F

    Optimisation Problem Using Derivatives

    Homework Statement a cylindrical tin can with volume 0.3l is being made, with the top and bottom sufaces twice the thickness as the sides. Show that a height to radius ration of h=4r will minimise the amount of aluminium required. Homework Equations V=\pi r^2 h \\ A = 2 \pi r^2 + 2 \pi r h...
  23. B

    Solve Optimal Factory Construct Problem for Max Goods in 10-50 Hours

    Hi all. I have a rather unique problem that I need help with and I really didn't know how to title it. Consider the following: You are trying to build a factory that produces the greatest amount of goods under a given time. You start off with a construction capacity of 20 per hour and you can...
  24. P

    MHB Tricky Production Function Optimisation

    Suppose that a firm in perfect competitive markets has the following production function: 1 1 Q = f (K,L) = K^(1/3)*L^(1/3) , where Q,K and L denote the production level, capital inputs and labour inputs. The firm's cost function is given by TC = r*K + w*L , where r is the costs of using each...
  25. dexterdev

    How can Raju effectively allocate his study time for his upcoming exam?

    This is a problem I got from my friend. I think this can be solved linear algebra. Please help me to transform this problem to mathematical equations at least. Raju has an exam after 1 month , and daily he allots 4 hours for study. He has 3 subjects to study (A, B and C) for the exam. Subject...
  26. G

    Nichrome wire heater optimisation

    Hi, I am designing a plastic recycler and I need advice on choosing gauge of some nichrome wire. I want the heater to reach around 220 degrees C and I will be powering it with a laptop charger (19v 7.1 amps). Temperature control will be achieved with a PID controller. The nichrome wire will...
  27. L

    Optimisation - Using the lagrange method

    Homework Statement The problem asks to design a cantilever beam of a minimum weight consisting of 2 steps. Given: total length (L), Force (F) at the end of the beam and allowable stress (σ) Need to find the diameters D and d, the length of the smaller shoulder of the beam (x)...
  28. L

    How to use the Legrange method to solve optimisation cantilever beam?

    esign a cantilever beam of minimum weight (volume) consisting of three steps that meets condition of strength. Given parameters are: total length of the beam L, force F at the end of the beam, and allowable stresses [sigma]. Parametres to be determined: Beam Diameters d1 d2 d3 Length x1...
  29. C

    Stuck with Optimisation question, help?

    S=8x2ln(1/2x) Find the value of x that gives a maximum. So far I have got, by differentiating: x2+ln(1/2x). [could be wrong] Btw the way in the question x is a ratio and so cannot equal zero. Please help and explain how to do it thanks :)
  30. J

    Optimisation Lagrangian Problem

    No this is not homework. http://imgur.com/zAZxmuC http://imgur.com/zAZxmuC Ok i am struggling to even start this question. I see it has a constraint so i would be tempted to use Lagrangian but from there i don't see how px and qy fit into it? Some assistance on the tools needed to...
  31. J

    Two-Variable Optimisation Confusion

    Hi, So f(x,y) = xe-x(y2 - 4y) Find all stationary points and classify them i got for fx(x,y) s.p (1,4),(1,0) for fy (x,y) s.p (0,2) I thought that you don't need double differentials at this stage and if it is a s.p it must satisfy for fx(x0,y0) = 0 for fy (x0,y0) = 0 which...
  32. P

    How Do I Solve This Multivariable Optimization Problem for Maximum Box Volume?

    Homework Statement I have 12m2 of cardboard. I must make a rectangular box with no lid, but I must maximize the potential volume of the box. The length, width and height of the box are x,y,z.Homework Equations I know that xyz = V (volume). I know the surface area is 12m2 ∴ 2xz+2yz+xy=12...
  33. G

    How Do You Calculate the Perimeter and Area of a Pontoon's Cross Section?

    Homework Statement This is the problem that I have , but I do not have any idea.. Please help me :) and sorry if it is not in the right category. Homework Equations The Attempt at a Solution
  34. E

    Optimizing Cost per Unit with Calculus

    Homework Statement REFER TO IMAGE Homework Equations SEE ABOVE The Attempt at a Solution I haven't been able to start this question. I'm wondering how to find the 'average cost per unit'.
  35. E

    What is the optimal volume of a box with given dimensions using calculus?

    Homework Statement SEE QUESTION IMAGE Homework Equations SEE ABOVE The Attempt at a Solution SEE WORKINGS IMAGE
  36. W

    ANSYS APDL/Workbench Optimisation

    I have a university coursework where I have to optimise the mass/volume of a bike frame using the cross-sectional area & tube thickness of the cylindrical tubing. I'm fairly competent in doing this process in ANSYS workbench (at least I'd like to think I am!). If I had the choice, I would use...
  37. S

    Recommended book for Optimisation?

    I'm looking to do a course on Optimisation, however there was no prescribed textbook and I'm a bit wary of doing a course without a textbook to reference. There was a generalised list given, of like 10 textbooks, but this is a bit too much, especially with 3 other subjects to do! Here is the...
  38. N

    How Do You Solve a Constrained Optimization Problem on a Unit Sphere?

    Homework Statement The temperature of a point on a unit sphere, centered at the origin, is given by T(x,y,y)=xy+yz Homework Equations I know that the equation of a unit sphere is x^2+y^2+x^2=1, which will be the constraint. The Attempt at a Solution The partial derivatives of T are...
  39. J

    Theoretical optimisation of foreign exchange strategy

    Hi everyone, I realize this is only loosely related to mathematics, but I didn't know where else to ask this question (if I post it in a finance forum I'll get 100 different answers from people who probably didn't really understand the question). I think mathematicians are the kind of people...
  40. D

    In Dire need of assistance for optimisation (Area/volume)

    Homework Statement A square-based rectangular prism has a total surface area of 54cm^2. Determine the side lengths if the volume is a maximum and hence find the volume. I have done a number of these and I am getting really annoyed because I always get all the sides being equal no matter...
  41. S

    Robust optimisation in Maths Programming

    I am a bit confused as to how to formulate the robust optimisation counterpart for the following problem, Homework Statement Consider the random linear constraint Ʃj ( ~aijxj ) ≤ bi, where ~aij's are the random parameters, Assume ~aij belongs to the uncertainty interval [aij-aij*, aij +...
  42. J

    Optimisation and best use of space

    Homework Statement I initially had to use the attached diagram to solve problems related to a concert venue.So I created a formula for perimeter and area and used these to create a formula for area with x as the only variable.I used differentiation to find the value of x when the area is at...
  43. V

    Optimisation along a curve on a surface

    Hi everyone, First of all I am not sure if I have chosen the right category for this posting but this looked the most reasonable out. I have a problem that I would like to solve but I am not sure where to look for answers. It seems like something other people might have worked with before...
  44. P

    How can I optimize 3000 variables for spline control points in image warping?

    Dear All Firstly thank you for looking at my post. I am trying to nonlinearly warp an image with respect to another using a free form deformation. I am trying to code the following paper in matlab: http://www.cs.jhu.edu/~cis/cista/746/papers/RueckertFreeFormBreastMRI.pdf I understand...
  45. J

    Optimise: Find Bearing to Intercept Enemy Jet in Shortest Time

    Homework Statement An enemy fighter jet has invaded friendly airspace and is traveling on a bearing of 40 degrees. At the time they scramble an interceptor from a nearby airbase, the enemy jet is 200km away and on a bearing of 10 degrees from the airbase. The enemy jet is traveling at...
  46. R

    Optimisation Question [Cylinder]

    Q. You have been asked to design a can shaped like a right circular cylinder with height h and radius r. Given that the can must hold exactly 130 cm3, what values of h and r will minimise the total surface area (including the top and bottom faces)? Volume(Cylinder) = pi(r)2*h 130 = pi(r)2*h...
  47. K

    Balsa Bridge Design Optimisation

    Hi, I need to design a balsa bridge that will cover a distance of 200mm. It will be loaded centrally for testing. The objective of this however is not to test until failure, but to have it fail at a certain weight. Points are also given for high strength to weight ratios and simplicity of the...
  48. L

    Optimisation using constraints

    Homework Statement Consider the intersection of two surfaces: an elliptic paraboloid z = x2 + 2x + 4y2 and a right circular cylinder x2 + y2 = 1. Use Lagrange multipliers to find the highest and lowest points on the curve of intersection The Attempt at a Solution L = x^2 + 2x + 4y^2...
  49. D

    Difficult Optimisation problem (maximizing a cuboid)

    Difficult Optimisation problem! (maximizing a cuboid) Find derivate d(x)
  50. A

    Minimizing Distance to Origin: Solving an Optimization Problem

    URGENT - optimisation problem Homework Statement Find the point on the line y=4x+7 that is closest to the origin. Homework Equations The Attempt at a Solution I am completely lost on where to start and what to minimise. It's for exam preparation!