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. H

    A Are these two optimization problems equivalent?

    Hello, I need help please. I have the following optimization problem defined as \begin{equation} \begin{aligned} & (\mathbf{P1}) \quad \max_{\mathbf{z}} \quad \left| d -\sum_{n=1}^{N} \frac{c_n}{f_n + z_n} \right|^2 \\ & \text{subject to} \quad \sum_{n=1}^{N} \frac{|a_n|^2 \text{Re}(z_n)}{|f_n...
  2. K

    A Parameter optimization for the eignevalues of a matrix

    Hello! I have a matrix (about 20 x 20), which corresponds to a given Hamiltonian. I would like to write an optimization code that matches the eigenvalues of this matrix to some experimentally measured energies. I wanted to use gradient descent, but that seems to not work in a straightforward...
  3. F

    A How can I group this data?

    I have a matrix of dimension 56*56, each row and column represent the compatibility of one person with the rest of the people. A sample matrix could be Alejandro Ana Beatriz Jose Juan Luz Maria Ruben...
  4. F

    I Optimization problem with multiple outputs: impossible?

    Hello, I'm facing a practical optimization problem for which I don't know whether a standard approach exists or not. I would have liked to rephrase the problem in a more general way, for the sake of "good math", but I'm afraid I would leave out some details that might be relevant. So, I'm going...
  5. Juanda

    Optimization of barrel length in pneumatic cannons

    I was checking bait cannons and potato guns on the internet because they are fun. Maybe one day I'll build my own. First of all, these cannons use multiple sources of energy (combustion using hair spray, dry ice, etc.). I'll just consider compressed air cannons because I think they are the most...
  6. T

    DeepMind AI Develops Efficient Sorting Algorithms

    https://arstechnica.com/science/2023/06/googles-deepmind-develops-a-system-that-writes-efficient-algorithms/
  7. codebpr

    A Finding a suitable form factor for a given set of conditions

    This is basically a physics problem but I will try my best to highlight the mathematics behind it. Suppose I have two functions: $$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$ $$\phi(z,B)=\int_0^z...
  8. tworitdash

    A Finding Global Minima in Likelihood Functions

    I have a likelihood function that has one global minima, but a lot of local ones too. I attach a figure with the likelihood function in 2D (it has two parameters). I have added a 3D view and a surface view of the likelihood function. I know there are many global optimizers that can be used to...
  9. B

    I Roulette system -- which optimization is better?

    Hey, gotta do some explanation first: I assume you know how roulette works. (if you dont: ball is thrown into a pit and it can either land on red, black or zero, each having a certain likeliness to land there. you can bet on where the ball will land) let's assume unrealistically you have the...
  10. R

    I Multivariable function optimization inconsistency

    Mentor note: For LaTeX here at this site, don't use single $ characters -- they don't work at all. See our LaTeX tutorial from the link at the lower left corner of the input text pane. I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the...
  11. SilverSoldier

    B Constrained Optimization with the KKT Approach

    I'm reading the book Deep Learning by Ian Goodfellow, Yoshua Bengio, and Aaron Courville, and currently reading this chapter on numerical methods--specifically, the section on constrained optimization. The book states the following. Suppose we wish to minimize a function...
  12. V

    Given an NLO reduce it to unconstrained optimization problem

    We are given the problem min x3-x42 such that (1): x12 + x3 = 2 and (2): (x2-x4)(x2+x4)=1. What I did was solve for x3 in (1) and then solve for x4 in (2). I substituted those equations into min x3-x42 and I obtain the solution: 2-x12-x22+1, would this be the correct approach to this problem...
  13. V

    Optimization Problem: x_1(sin(x_1)) such that exp(x_1)-1>=0

    I know to solve this problem we need to see if x1sinx1 is convex and if the constraint is convex. I already know that x1sinx1 is not convex so the problem is not convex, but for proving that this function is not convex is where I am confused. But how do I go about showing this? I'm assuming I...
  14. V

    Continuous Optimization, is this convex?

    f(x)=ln(|x1|+1)+(-2x1 2 +3x2 2 + 2x3 3) + sin(x1 + x2 + x3), for this problem in particular would be it be sufficient to find the Hessian and to see if that matrix is semi positive definite to determine if it convex?
  15. Keysa

    How to find the positive maximum value of a function

    This is the code that i wrote Clear["Global`*"] Z = 500; W = 100000; G = 250; H = 100; K = 0.5; T = 30; L = 4000; P = 5; S = 2.5; Y = 1; A = 0.1; V = 2.5; J = 8000; f[x_] := 1/ x {(J*Z*x*(2*Y - x))/( 2*Y) - ((W + T*G) + ((L + T*P)*2*Z*Y*(1 - ((Y - x)/Y)^1.5))/ 3 + (H + T*S +...
  16. F

    Stationary points classification using definiteness of the Lagrangian

    Hello, I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse. So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
  17. B

    Does anyone know about axis-symmetric topology optimization?

    ABAQUS provides geometric restrictions such as a planer, rotational, and other symmetric, but there is no axis symmetric restriction. I know that the 2D axis symmetric element model could be possible to make in PART section. But I want to know that a full 3D element model could be optimized by...
  18. M

    Optimization: Dual for L1 norm minimization with equality constraint

    Hi, I was reading through some notes on standard problems and their corresponding dual problems. I came across the L2 norm minimization for an equality constraint, and then I thought how one might formulate the dual problem if we had an L1-norm instead. Question: Consider the following...
  19. M

    Convex Optimization: Dual Function Definition

    Hi, I was working through the following problem and I am getting confused with the solution's definition of the dual. Problem: Given the optimization problem: minimize ## x^2 + 1 ## s.t. ## (x - 2) (x - 4) \leq 0 ## Attempt: I can define the Lagrangian as: L(x, \lambda) = (x^2 + 1) + \lambda...
  20. B

    Bayesian Optimization Theory

    Hello, I am better studying the theory that is the basis of Bayesian optimization with a Gaussian Process and the acquisition function EI. I would like to expose what I think I understand and ask you to correct me if I'm wrong. The aim is to find the best ##\theta## parameters for a parametric...
  21. shivajikobardan

    MHB Which 2 exams should I skip for best optimization for learning?

    https://lh5.googleusercontent.com/aEaRfwwvnXOSUT8p390UKsIjrbVi2ERlBaKjwUDR3JoQOEw8bCLhQIek9wPo83GrJ8wqG7WTC1p1eQRgPdap9cPx9gt8zCQnibrb6BQjyvYVS91m2c79diOIVqHeKG0uSIo6phoT So I messed up in a tactical move. I was studying artificial intelligence for my backlog exam(exam that I failed also called...
  22. shivajikobardan

    Testing Which 2 exams should I skip for best optimization of learning?

    So I messed up in a tactical move. I was studying artificial intelligence for my backlog exam(exam that I failed also called supplementary exam, re-exam, retaking exam). That was so huge and due to lots of other reasons (I don't want to sound whining so not mentioning them), I am here. I have...
  23. Dario56

    I What Exactly is Step Size in Gradient Descent Method?

    Gradient descent is numerical optimization method for finding local/global minimum of function. It is given by following formula: $$ x_{n+1} = x_n - \alpha \nabla f(x_n) $$ There is countless content on internet about this method use in machine learning. However, there is one thing I don't...
  24. M

    Optimization: Formulation of the dual of a semi-definite program (SDP)

    Hi, I was working through the following optimization problem, and am getting stuck on how to get to the dual problem that is being presented. Question: Find the dual problem for the semidefinite primal problem below: min_{X} tr(C^T X) \text{subject to} AX = B X \succeq 0 (the answer is...
  25. M

    Optimization: How to find the dual problem?

    Hi, I am working on the following optimization problem, and am confused how to apply the Lagrangian in the following scenario: Question: Let us look at the following problem \min_{x \in \mathbb{R}_{+} ^{n}} \sum_{j=1}^{m} x_j log(x_j) \text{subject to} A \vec{x} \leq b \rightarrow A\vec{x}...
  26. A

    I Help with rewriting a compound inequality

    See attached screenshot. Stumped on this, I'll take anything at this point (hints, solution, etc).
  27. F

    Optimization Problem - Dynamic Programming

    Summary:: Hi, this is an exercise from an algorithm course. I have been trying for hours but I have no successful ideas on how to solve it. I can only understand that DP is the correct approach, since Greedy method does not work. Suppose you have *n* friends that wants to give you an amount of...
  28. P

    A Proving the Equivalence of Local and Global Maxima for Concave Functions

    Consider the following theorem: Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.## The intuition is that local maxima and global maxima coincide for...
  29. C

    PDB file using GAUSSIAN 16 optimization

    Hello There, I am studying docking of ligand molecules into DNA using Autodock Vina. Before doing that I optimize the ligand molecule using Gaussian 16. I want to know how can we get the PDB format after optimization. Do I just need to open the .chk file and save it as PDB format or do I need...
  30. M

    MHB Optimization - Lagrange multipliers : minimum cost/maximum production

    Hey! :giggle: Business operates on the basis of the production function $Q=25\cdot K^{1/3}\cdot L^{2/3}$ (where $L$ = units of work and $K$ = units of capital). If the prices of inputs $K$ and $L$ are respectively $3$ euros and $6$ euros per unit, then find : a) the optimal combination of...
  31. maistral

    A Optimization with integers as results

    Say for example I have a dataset (X, Y) which I need to fit to the function y = Ax^2 + By + Cxy. How do I retrieve values of A, B, and C such that they can only be integers? As of now I'm doing grid search which is so taxing.
  32. person123

    I Fitting Data to Grafted Distribution

    I have a set of data (representing the strength distribution of samples), and I would like to fit a normal-Weibull grafted distribution. To the left of a specified graft point, the distribution is Weibull, and to the right it's normal. At the graft point, the value and the first derivative are...
  33. M

    Condenser optimization question

    Hi, I have an attempt at a plate heat exchanger (condenser) that uses water to condenser refrigerant, as a part of a heat pump. I have a total heat load of 12.01 kW. My current heat load is 10 kW. I have an analytical error on the wall temperature of about 23%, if I use Excel's Solver to...
  34. Leo Liu

    Optimization with Lagrangian Multipliers

    Problem: Solution: My question: My reasoning was that if x is max at the point then the gradient vector of g at the point has only x component; that is ##g_y=0,\, g_z=0##. This way I got: $$\begin{cases} 4y^3+x+z=0\\ \\ 4z^3+x+y=0\\ \\ \underbrace{x^4+y^4+z^4+xy+yz+zx=6}_\text{constraint...
  35. rxh140630

    Optimization problem - right circular cylinder inscribed in cone

    Please I do not want the answer, I just want understanding as to why my logic is faulty. Included as an attachment is how I picture the problem. My logic: Take the volume of the cone, subtract it by the volume of the cylinder. Take the derivative. from here I can find the point that the cone...
  36. L

    I Frechet Derivatives & Optimization - Mechanics Example

    Allegedly Frechet derivatives are used in optimization problems in mechanics, but I have not found a clear example of this. Does anyone know of an example to go through? I would think because of the significance of Lagrangian mechanics that it could be more related to a variational calculus...
  37. U

    MHB How to solve following optimization problem?

    The following is the mathematical expression for my model's rate expression. Variables $x,y$ are the controlling parameter, while the rest are positive constants. $$\max_{x,y} \ ax + by^3 \ (s.t. \ 0\leq x \leq 1,\ 0\leq y\leq1)$$ Can I mathematically say that it is a convex problem within...
  38. J

    Fortran GFORTRAN Optimization Question

    I am trying to troubleshoot why GFORTRAN versions beyond 5.4 will not compile with optimization on some of my .f source. You can request options included in each level by: Gfortran -Q -O1 --help=optimizers > listO1.txt (as an example) When I enter the enabled flags individually and compile...
  39. D

    I Multivariable optimization problem

    Hi all, (Please move to general or mechanical engineering sub-forum if more appropriate over there. I put this here as it is essentially a mathematics problem.) Broken into sections: - problem categorization (what type of problem I think I have), - the question, - specifics (description of the...
  40. anemone

    MHB Max & Min Values of $S$ for $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$

    Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.
  41. archaic

    Java JavaFX layout not updating and email sending optimization problem

    I am writing a java application that would let me bulk send emails. The first problem I have is that of performance; approximately 15 seconds per 5 emails. The second problem, which is the more important, is that my JavaFX is not updating the scene. My code below shows that the way I intended...
  42. A

    A Differential Equations (Control Optimization Problem)

    \begin{equation} y_{1}{}'=y_1{}+y_{2} \end{equation} \begin{equation} y_{2}{}'=y_2{}+u \end{equation} build a control \begin{equation} u \epsilon L^{2} (0,1) \end{equation} for the care of the appropriate system solution \begin{equation} y_{1}(0)=y_{2}(0)=0 \end{equation} satisfy...
  43. Saptarshi Sarkar

    Optimization of the distance from the point on an ellipse

    My Attempt :We need to maximize ## D=\sqrt{x^2+(y+2)^2} ## subject to the constraint ##4x^2 + 5y^2 = 20##. From the constraint equation, we can write ##x^2=\frac{20-5y^2}{4}## Using this in the formula for distance, ##D=\sqrt{\frac{20-5y^2}{4}+(y+2)^2}## Differentiating this wrt y, and...
  44. H

    Linear optimization problem

    attached resolution attempt
  45. Hiero

    I Optimization of multiple integrals

    The Euler Lagrange equation finds functions ##x_i(t)## which optimizes the definite integral ##\int L(x_i(t),\dot x_i(t))dt## Is there any extensions of this to multiple integrals? How do we optimize ##\int \int \int L(x(t,u,v),\dot x(t,u,v))dtdudv## ? In particular I was curious to try to...
  46. fresh_42

    I QM through stochastic optimization on spacetimes

    I have a simple question as a layman in the field: Is this worth reading, and even more, is it a contribution to possibly shorten the endless discussions in this subforum? https://www.nature.com/articles/s41598-019-56357-3.pdf
  47. Quantum computing & chill

    Quantum computing & chill

    A thing doing its own thingy thing could compute faster than a computer can compute.
  48. W

    MHB Optimization calculus question (Difficult)

    A truck crossing the prairies at constant speed of 110km per hour gets 8km per litre of gas. Gas costs 0.68 dollars per litre. The truck loses 0.10 km per litre in fuel efficiency for each km per hour increase in speed. Drivers are paid 35 dollars per hour in wages benefits. Fixed costs for...
  49. Mathman2013

    Optimization problem (Max area of a combined semi circle and a square)

    A figure is made from a semi circle and square. With the following dimensions, width = w, and length = l. Find the maximum area when the combined perimiter is 8 meter. I first try to construct the a function for the perimeter. 2*l + w + 22/7*w/2 = 8 - > l = 4 - (9*w)/7 Next I insert this...
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