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. M

    MHB Extrapolation and interpolation in line search optimization

    hi can you tell me these equations: A = 6*(f2-f3)/z3+3*(d2+d3); % cubic fit B = 3*(f3-f2)-z3*(d3+2*d2); z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A; % numerical error in MATLAB fmincg.m...
  2. I

    I Discrete Optimization Problem?

    Consider the expression:$$A = \frac{ M! }{ r_1!\ r_2! }$$ where M = r_1 + r_2 , where r_1 = (M - 2r_2) $$A = \frac{ (r_1 + r_2)! }{ r_1!\ r_2! } \\ \ \\ \ = \frac{ ((M-2r_2) + r_2)! }{ (M-2r_2)!\ (r_2)! } \\ \ \\ \ = \frac{ (M-r_2)! }{ (M-2r_2)!\ r_2! } $$ Then, for a...
  3. mertcan

    Convex Optimization Without Slater Condition

    Hi, initially I am aware of the fact that when slater condition holds, then dual optimum equals primal optimum in convex optimization. But if slater condition does not hold then dual gap exist. When we have nonlinear nonconvex optimization we apply convexification of constraints including...
  4. N

    A Particle swarm optimization for matrix inversion

    Hi everyone, I am working on matrix inversion and focusing on low-complexity method such as iterative method. Recently, I am interested to explore how particle swarm optimization (PSO) can be applied to do matrix inversion. Since I am very very new in PSO, I have no idea how to start my work...
  5. F

    Can someone recommend an algorithm to optimize this data

    I have a data set of samples, and I made up some variables that filter out unwanted samples from the data set. Say each sample has values ##\vec{x}=x_1, x_2,..., x_n##, and I know the values of ##\vec{x}## for each sample. Also, if I sum up all of the samples, I get a value, ##Z##, which tells...
  6. F

    Moving a particle from point A to B on a efficient way in GAMS

    Hello everyone. I am triyng to calculate the route which takes less time to go from point A to point B in the presence of a constant flow (I. E. a simple version of Zermelo's navigation problem) using the GAMS software. However, if I put both points on a straight line and make the constant flow...
  7. Specter

    Gr 12 Calc: Optimization problem/min cost

    Homework Statement A homeowner wishes to enclose a rectangular garden with fencing. The garden will be adjacent to his neighbour’s lot. There will be fencing on all four sides. His neighbour will be paying for half the shared fence What should the dimensions of the garden be if the area is...
  8. CK_KoopaTroopa

    I Finding the max angle of a longboard deck before the wheels slip

    Hi, I'm making an electric longboard and trying to write an app for my phone to function as the remote. I've got a bunch of fancy Star-Trek-esque indicators on it, one of which is the pitch and roll of the deck. All the indicators have "danger zones" and turn red when they hit them, and for this...
  9. TeeTex

    Calculus Book about optimization problems

    hi guys, i am preparing my self for the calculus 1 2 3 final and i need recommendation about optimization problems theories book or something to help me understand how to solve and understand optimization problems and to solve them. Thanks♥
  10. T

    Optimization lagrangian problem

    Homework Statement I would like to solve for Y an optimisation problem Homework Equations Max Y'C + Y'Br + αr0 Subject to : k=sqrt(Y'ΣY) Y'e + α = 1 Where Y, C and B are columns vector of n lines. Σ is symetric matrix of n order e =(1,...1)' and α is a reel parameter. I did calculus with...
  11. C

    A Combinatorial optimization problem

    Hi, I have the following optimization problem. I have a list of tasks that I should be able to perform with my tools. Each tool costs a certain amount of money, and may be used to carry out a finite number of tasks. The goal is to choose an optimal set of tools in such a way that the toolset can...
  12. Guy Fieri

    Issue With Optimization Problem

    Homework Statement Homework Equations I have yet to figure out any relevant equations, but I do believe that the constraint equation for the optimization problem is the y=64-x^6 listed above. The Attempt at a Solution I am currently trying to figure out methods to begin my optimization...
  13. Brandon Preble

    Trebuchet Throwing Arm Optimization

    Summary: Some help would be greatly appreciated for finding the ideal taper and x/y dimensions of a trebuchet throwing arm to optimize strength and minimize the moment of inertia. Long Version: I am a high school junior currently in the process of working on an ambitious project to build a 21...
  14. synMehdi

    A Pontryagin minimum principle with control constraints

    Hi, I am trying to solve a control problem where I have to minimize the fuel consumption of a vehicle: $$J=\int_{0}^{T} L(x(t), u(t),t) + g(x(T),T)dt$$ ##L(u(t),v(t))=\sum\limits_{i,j=0}^{2} K_{i,j} u(t)^i v(t)^j ## is convex (quadratic) and the term ##g(x(T),T)## is to have a constraint in the...
  15. R

    Creating system of equations from word problem optimization

    I have this word problem, and was wondering how I would go about creating a system of equations. Here is the question: Problem: You are a small forest landowner, and decide you want to sustainably harvest some of timber on your property. There are costs related to the infrastructure needed to...
  16. F

    Optimization with Newton's method

    I have a system of equations which I solved with Newtons method. Call Newtons method a function NM=f(K1,K2). K1 and K2 are input and a vector of x=x1,x2,x3,x4 is output. I have another function, SSR, the sum of square residuals. It looks like this : $$\sum...
  17. SSGD

    I Optimization of an Amplifer Circuit

    The gains of an OP-AMP are listed below: G_d = (R_1*R_4+R_2*R_3+2*R_2*R_4)/(2*R_1*(R_3+R_4)) G_s = (R_1*R_4-R_2*R_3)/(R_1*(R_3+R_4)) \frac {\partial G_d} {\partial R_1} = -R_2*(R_3+2*R_4)/(2*R_1^2*(R_3+R_4)) My questions is... Is there a mathematical perform the following: Differential gain...
  18. F

    A Optimization of a nonlinear system

    Can someone please tell me how to go about optimizing this system of equations? It is weird because the residuals are computed with ##A = B*X_1+C*X_2## but X_1 and X_2 are computed in a separate function ##[X_1,X_2]=f(k1,k2,H0,G0)##, and what I am optimizing is a parameter in that second...
  19. kandelabr

    Trajectory with minimum acceleration

    Currently design of turbomachinery (impellers/turbines) is more a form of art than an engineering process. One has to guess a bunch of parameters and check if they are right in much later stages of design. I was thinking about designing the other way: we know the initial and the final velocity...
  20. FallenApple

    Optimization Problem with a Constraint

    Homework Statement This is a leetcode question. You are a professional robber planning to rob houses along a street. Each house has a certain amount of money stashed, the only constraint stopping you from robbing each of them is that adjacent houses have security system connected and it will...
  21. U

    Blades/propellers - power optimization - drone hovering

    I wonder what is the best figures for blades/prepellers - assuming I am trying to optimize battery life and my main usage for the drone is hoovering. I can not play with the arm length. I do consider 4 blades over 2 blades at the same propeller - not sure if it will help. Any feedback or idea is...
  22. Mr Davis 97

    I Maximizing xy: Understanding Optimization Problems in Mathematics

    I am little confused when it comes to optimization problems. For example, say we are given that ##x+y=2##, and are asked to maximize ##xy##. By AM-GM, we have that ##xy \le 1##. But why should this indicate that ##1## is the maximum value? Isn't it an equally true statement to claim that ##xy...
  23. Mr Davis 97

    Maximize xy when x + y + z = 3

    Homework Statement For positive x, y, z where ##x \le y \le z## such that x + y + z = 3, what is the maximum value of ##xy##? Homework EquationsThe Attempt at a Solution First, before I attempt a solution, isn't it the case that ##x=y=z=1##, since the only partition of 3 into three terms is...
  24. Q

    Rectangle optimization - possible text error

    Hi, I may have discovered a textbook error but I'm no calc whiz. I need an assist to find out if the question unintentionally described a square instead of a rectangle. I have attached the textbooks solution as well as my attempt at a solution. The numbers check out, I just want to make sure...
  25. 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...
  26. Jamison Lahman

    Algorithm Optimization [Python]

    Homework Statement Given a list of integers and a single sum value, return the first two values (parse from the left please) in order of appearance that add up to form the sum. sum_pairs([11, 3, 7, 5], 10) # ^--^ 3 + 7 = 10 == [3, 7] sum_pairs([4, 3, 2, 3, 4]...
  27. S

    Counterexamples to Claim that A = (10,1,1,10) for B = (10,1,1,10)?

    I'm trying to solve a problem that amounts to: Given b0, ..., bn-1 where1 <= bi, find the max of |a0 - a1| + |a1 - a2| + ... + |an-2 - an-1| where 1 <= ai <= bi. I'm 100% confident that each ai is either 1 or bi. I'm 90% confident that the elements a0, ..., an-1 are either 1, b0, 1, b1...
  28. danielhep

    Optimization w/ Constraint Question (Multivariable Calculus)

    Homework Statement Find any maxima/minima on f(x,y) = x2+2y2 on the unit circle, centered at the origin. Homework Equations grad f = λgrad g constraint: 1=x2+y2 The Attempt at a Solution grad f = 2xi+4yj grad g = 2xi+2yj 2x=λ2x 2y=λ4y How do I solve this? I don't see any way to get numbers...
  29. Z

    MHB Maximize City Kid Earnings: Peas & Carrots

    I am a city kid with a windowsill with the dimensions 12x5 inches. I have peas which require a 1x4 space and make me 15 dollars, and carrots which require 1x5 space and make me 50. I can only make 15 sets of peas if I were to use the whole space, and 12 sets of carrots if I were to use the whole...
  30. A

    Rienforced Concrete Column Cost Optimization with Genetic Algorithm

    Hi, I'm a Civil Engineering student doing my final project for my BE degree. I have developed a software that uses a genetic algorithm for optimizing reinforced concrete columns. I have tested my program with a 17 stories building, and got some very unusual results in the design but much...
  31. B

    Producing a Family of 0,1 Knapsack Sets

    Dear Physics Forum friends, I am currently stuck with the following question about the integer optimization: "Produce a family of 0,1 knapsack sets (having an increasing number n of variables) whose associated family of minimal covers grows exponentially with n." My thought is that I need to...
  32. Mechanic7

    Hydraulic system - Design optimization

    Homework Statement Hello all, I'm a Mechanical engineer with no prior experience in designing flow systems. I'm struggling with a hydraulic system that I have to design. the number of unknowns is large. but maybe by dividing the system to small sections, a solution can be found. I attached...
  33. Mechanic7

    Water flow system - Design an Optimization

    Hello all, I'm a Mechanical engineer with no prior experience in designing flow systems. I'm struggling with a hydraulic system that I have to design. the number of unknowns is large. but maybe by dividing the system to small sections, a solution can be found. I attached a schematic drawing...
  34. M

    MHB Linear Optimization: Possibilities

    Hey! :o To product a product $3$ square plates $A$ with measures $300\times 500$mm and $2$ square plates $B$ with measures $400\times 600$mm are needed. These plates must be cut out by two different workpieces with measres $1000\times 1000$mm (Type I) and $600\times 1500$mm (Type II). The...
  35. T

    Multi-criteria optimization. How to solve it?

    http://tinypic.com/r/1570ojk/9 Please see the image. I don't understand how to solve such a problem. It has three criteria. Any hints or guide would be appreciated or even a solution. This is actually just an example from the book, but the book doesn't even solve the question, so I don't even...
  36. E

    Sparsity of Support vector machines over an RKHS

    Im trying to solve the following problem from the book 'Learning with kernels', and would really appreciate a little help. Background information - Let $\{(x_{1},y_{1}),...,(x_{N},y_{N})\}$ be a dataset, L a Loss function and $H(k)$ a reproducing kernel Hilbert space with kernel $k$. The...
  37. T

    How to properly use the nlm function in R for minimizing functions?

    I try to find out how to minimize functions i R by using nlm function: > f<-function(x,y){x^2+y^2+10-5*x-y} > nlm(f,0.1,0.1) That only gives me an estimate for x. How would write the code to get x and y?
  38. C

    Is there a worked-out example of L-BFGS / L-BFGS-B?

    I have seen the implementation of L-BFGS-B by authors in Fortran and ports in several languages. I am trying to implement the algorithm on my own. I am having difficulty grasping a few steps. Is there a worked out example using L-BFGS or L-BFGS-B ? Something similar to...
  39. J

    A Optimization problem classification

    Here all, Here's a problem I'm trying to solve. Given a planar piecewise linear and circular curve (ie. a curve consisting of line and circular arc segments) that represents the path of a particle; a set of rules for traversing the two types of curve segments as well as the transitions between...
  40. BiGyElLoWhAt

    Optimization Problem with an RC BP filter

    I am assigned to design a circuit that peaks voltage at 10kHz and is less than half peak voltage at 3k and 30k. Only capacitors and resistors are allowed. The circuit I'm using is attached. I end up with ##I_0 = [\frac{-R_2\omega^2C_1C_2 + [C_1+C_2]R_1R_2\omega^3C_1C_2 - R_1\omega^2[C_1+C_2]^2...
  41. T

    I Dot product constrained optimization

    Problem: Fix some vector ##\vec{a} \in R^n \setminus \vec{0}## and define ##f( \vec{x} ) = \vec{a} \cdot \vec{x}##. Give an expression for the maximum of ##f(\vec{x})## subject to ##||\vec{x}||_2 = 1##. My work: Seems like a lagrange multiplier problem. I have ##\mathcal{L}(\vec{x},\lambda)...
  42. F

    I Optimization in Projectile Motion

    Dear Forum, I am working on projectile motion and optimization these days. Projectile motion uses the kinematic equations of free fall for the vertical direction together with the equations of constant velocity in the horizontal direction. Maximum Range: if an object is launched at a certain...
  43. P

    Linear Optimization Conversion (nested free variable)

    Homework Statement (This is a mock-up) Given maximize(x,y) Ax + | x - y | s.t. x + By - C >= 0 y >= 0 A, B, and C are some coefficients. Homework Equations Linearizing absolute expressions: z = | x |; where z >= x, z >= -x Linearizing free variables: let x = x+ - x-; where x+, x- >= 0...
  44. D

    A Gaussian 09 Output file_DFT Calcuation

    Hi, I am trying to understand the functional form of B3LYP from the Gaussian output file. I have tried to relate the details in the output file with the functional form of B3LYP. But I am not sure what certain terms correspond to. I have mentioned below the details. Can you pleaese help me. In...
  45. S

    A Can this optimization problem be solved?

    Hello, I am working on an optimization problem but I am not sure if the problem can be formulated and solved with conventional solvers. Assume the minimization problem for a set of elements ##\mathcal{N} =\{ 1,\dots, h, \dots, i,\dots, N \}## $$ \mathrm{minimize}\quad C = \sum_{i=1}^{N}...
  46. kolleamm

    C# Search optimization for if statements in C#

    I have a program that searches out string types.. Here is what I have so far - C# language. if(word.Contains "Hello") {...} else if(word.Contains "Bye") {...} else if(word.Contains "Yes") {...} ... I plan to add more and more of those if statements as my program advances. Would organizing...
  47. K

    MHB How Does Changing the Sign of a Constraint Affect Lagrangian Solutions?

    A student wishes to minimize the time required to gain a given expected average grade, 𝑚, in her end-of-semester examinations. Let {t}_{i} be the time spent studying subject i\in{1,2}. Suppose that the expected grade functions are {g}_{1}({t}_{1}) = 40+8\sqrt{{t}_{i}} and {g}_{2}({t}_{2}) =...
  48. Kefeng

    Question about CMA-ES step size sigma

    Hi everyone, I am new here. I am working in geophysics and I would like to invert for a simple layered velocity model using CMA-ES optimization method. I downloaded the purecmaes.m code in Matlab here: https://www.lri.fr/~hansen/cmaes_inmatlab.html, and also implemented one in Fortran 90. I...
  49. E

    A Maximizing Two Functions with Constraints in Phase Covariant Cloning Machine

    hello, in the task of finding the optimal phase covariant cloning machine, i have to maximize two functions of six variables :f1=a.C+b.D and f2=a.B+c.D , they are many constraints, but I've already used them to get to those expressions in the first place, the variables are real scalars and vary...
  50. Alex Sieber

    Optimizing Force of Solenoid w/Dia. ≤ 15mm

    First time posting here so excuse me if I don't know the rules so well. I figured this would be the best place to post this question. I'm trying to optimize the force produced by a solenoid that is no bigger than 15mm in diameter (D). My goal is to get just the right balance of number of wire...
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