Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    Only €10,99/month after trial. Cancel anytime.

    Mathematical Optimization: Fundamentals and Applications
    Mathematical Optimization: Fundamentals and Applications
    Mathematical Optimization: Fundamentals and Applications
    Ebook150 pages1 hour

    Mathematical Optimization: Fundamentals and Applications

    By Fouad Sabry

    ()

    Read preview

    About this ebook

    What Is Mathematical Optimization


    Mathematical optimization, often known as mathematical programming, is the process of choosing, from among a group of potential solutions, one that is optimal with relation to a set of predetermined criteria. Discrete optimization and continuous optimization are the two subfields that make up the majority of this field. Problems related to optimization appear in each and every one of the quantitative subfields, from computer science and engineering to operations research and economics. For millennia, the field of mathematics has been interested in the creation of methods that may solve these problems.


    How You Will Benefit


    (I) Insights, and validations about the following topics:


    Chapter 1: Mathematical optimization


    Chapter 2: Brachistochrone curve


    Chapter 3: Curve fitting


    Chapter 4: Deterministic global optimization


    Chapter 5: Goal programming


    Chapter 6: Least squares


    Chapter 7: Process optimization


    Chapter 8: Simulation-based optimization


    Chapter 9: Calculus of variations


    Chapter 10: Vehicle routing problem


    (II) Answering the public top questions about mathematical optimization.


    (III) Real world examples for the usage of mathematical optimization in many fields.


    (IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of mathematical optimization' technologies.


    Who This Book Is For


    Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of mathematical optimization.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateJun 30, 2023
    Mathematical Optimization: Fundamentals and Applications

    Read more from Fouad Sabry

    Related to Mathematical Optimization

    Titles in the series (100)

    View More
    View More

    Related ebooks

    Intelligence (AI) & Semantics For You

    View More
    View More

    Related podcast episodes

    Related articles

    • Article

      Microcontrollers In Amateur Radio

      Aug 1, 2022

      My way of teaching about program data has always been a little different than the way most approach the subject. As you may know, pointers in C are a special type of variable that allows you to access data in a very efficient manner. Indeed, many com

      6 min read

    • Article

      Expect And Unwrap

      May 31, 2022

      Both expect() and unwrap() are used with the Result and Option data types and allow you to deal with error and unexpected conditions. Put simply, they allow you to get the value from a Result or Option variable, if it exists, or deal with the error c

      1 min read

    • Article

      Expect And Unwrap

      May 31, 2022

      Both expect() and unwrap() are used with the Result and Option data types and allow you to deal with error and unexpected conditions. Put simply, they allow you to get the value from a Result or Option variable, if it exists, or deal with the error c

      1 min read

    • Article

      Test If You’re Human With Our PHP Captcha

      Nov 12, 2024

      Creating a captcha is something of a challenge; you need to balance it being too difficult for a computer program to solve against not being so hard that a human can’t pass its test. Most of us have probably experienced difficult captchas that featur

      9 min read

    • Article

      How To Solve Mazes Using Go And Maths

      May 5, 2020

      This month, we’re going delve into solving maze and dungeon puzzles with the help of graph theory (well, we are all in isolation–Ed). The purpose of such puzzles is to determine whether there’s a path that connects two places and find that path, keep

      7 min read

    • Article

      PYTHON/GO Parsing XML files

      Jul 2, 2019

      PYTHON/GO Parsing XML files
      8 min read

    • Article

      Adding Kevlar® to Your Code

      Feb 1, 2022

      Someone once said that writing the code for a new program takes about 20% of the development time. Testing and debugging that code eats up the remaining 80%. Kinda makes you wonder how programmers manage to screw up the first 20% to such a degree tha

      4 min read

    • Article

      Discover The All-new V Language

      Apr 5, 2022

      Credit: https://vlang.io No, V isn’t a fan fiction release of the famous 1980s TV series. Instead, it’s a fast, modern, open source, statically typed compiled programming language. V can be used in various areas including web development, systems pro

      9 min read

    • Article

      Discover The All-new V Language

      Apr 5, 2022

      Credit: https://vlang.io No, V isn’t a fan fiction release of the famous 1980s TV series. Instead, it’s a fast, modern, open source, statically typed compiled programming language. V can be used in various areas including web development, systems pro

      9 min read

    • Article

      Spice Up Your Python Console Applications

      May 30, 2023

      Credit: https://github.com/Textualize/rich Matt Holder has been a fan of the open source methodology for over two decades and uses Linux and other tools where possible. In his spare time, he enjoys listening to music and reading. When writing termina

      5 min read

    • Article

      Creating Functions

      Jul 31, 2019

      Creating Functions
      2 min read

    • Article

      Code Read/write System File Tools

      May 31, 2022

      The subject of this tutorial is file input and output (I/O) in Rust. File I/O is an important part of every operating system. An OS or even a database system wouldn’t be able to function without being able to process, read, write and append to files.

      8 min read

    • Article

      Visualising Experimental Results

      Mar 10, 2020

      All the examples in the main part of this tutorial generated results purely within LibreOffice Calc, using the laws of maths. However, it is also useful for creating graphs of experimental results, perhaps combined with equivalent results from simula

      1 min read

    • Article

      Fix LEDs With Some Coded CAD Models

      Mar 5, 2024

      Don’t miss next issue, subscribe on page 16! Tam Hanna’s creativity can be compared to the amount of humour found in a stone or the communication capability of popped balloon. In short, he is the ideal candidate to explain how to design cool things u

      7 min read

    • Article

      Enhance GIMP With Must-have Plugins

      Apr 4, 2023

      Credit: http://gimp.org Karsten Günther loves to extend his GIMP installations with useful plugins. And playing with the new tools… Many of the most important plugins presented here are quite old, but they are still maintained and sometimes even deve

      9 min read

    • Article

      Visualise Complex Data In Style Using Timelion

      Oct 20, 2020

      Simon Quain is a site reliability engineer who likes discovering open datasets online to play around with in the Elastic Stack. You’ve probably heard of Elasticsearch – the search engine that enables you to index and then quickly search through your

      9 min read

    • Article

      Plot Geolocation Data With ELK

      Dec 15, 2020

      Simon Quain works as a site reliability engineer who likes finding open datasets online to play around with in the Elastic Stack. If you need to increase the resources available to Logstash or Elasticsearch, edit the -Xms and and -Xmx parameters in

      8 min read

    • Article

      Chatty AI man

      Jun 27, 2023

      At the time of writing, the official way to get access to the model data involves visiting A https://bit.ly/lxf304form and filling out the form. Sadly, practical experience teaches that non-edu email addresses usually don’t get a positive result. The

      4 min read

    • Article

      GO Inside Parsing – How Go Handles The Code

      Jul 30, 2019

      This tutorial has two aspects: a theoretical one and a practical one. In the theoretical part, you will learn about parsing, grammar and regular expressions; this is how languages are built and therefore understood in terms of construction and usage.

      8 min read

    • Article

      Get Started With PCalc

      Apr 19, 2021

      Get Started With PCalc
      2 min read

    • Article

      Solve Word Puzzles With Clever Code

      Apr 2, 2024

      Matt Holder is an IT professional of 15 years, Linux user for over 20 years, homeautomation fan and selfprofessed geek. The full source code can be downloaded from https://github.com/mattmole/LXF-Countdown-Word-Solver We are going to create a program

      8 min read

    Related categories

    Reviews for Mathematical Optimization

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      Mathematical Optimization - Fouad Sabry

      Chapter 1: Mathematical optimization

      The selection of a best element, with reference to some criteria, from some collection of accessible alternatives is the goal of mathematical programming as well as mathematical optimization, which may also be written as the alternate spelling optimisation.

      An optimization issue is defined under the more general approach as the process of either maximizing or reducing the value of a real function. This is accomplished by methodically selecting input values from within an acceptable set and calculating the value of the function. A significant portion of the field of applied mathematics is dedicated to the application of optimization theory and methods to many different formulations. In a broader sense, optimization refers to the process of determining the best available values of a particular objective function when that function is applied to a defined domain (or input). This process can be applied to a wide variety of different types of objective functions and a wide variety of different types of domains.

      Depending on whether the variables being optimized are continuous or discrete, optimization issues may be broken down into two distinct types:

      A discrete optimization problem is an optimization problem that has discrete variables. In a discrete optimization problem, an object such an integer, permutation, or graph has to be located from a set that has a countable size.

      The process of extracting an optimum value from a continuous function is referred to as continuous optimization, and it is used to describe a kind of issue that involves continuous variables. Constrained difficulties and multimodal problems are two examples of these types of issues.

      The following is one possible representation of a problem requiring optimization::

      Given: a function f : A → ℝ from some set A to the real numbers

      Sought: an element x0 ∈ A such that f(x0) ≤ f(x) for all x ∈ A (minimization) or such that f(x0) ≥ f(x) for all x ∈ A (maximization).

      A issue with such a formulation is referred to as an optimization problem or a mathematical programming problem (a phrase that is not directly connected to computer programming but is still in use, for example in linear programming; see the section on History below for more information). This overarching paradigm lends itself well to the modeling of a wide variety of theoretical and practical challenges.

      Considering that the following is true

      {\displaystyle f(\mathbf {x} _{0})\geq f(\mathbf {x} )\Leftrightarrow -f(\mathbf {x} _{0})\leq -f(\mathbf {x} ),}

      It is sufficient to merely address issues involving minimization. On the other hand, the alternative viewpoint, which is to address solely situations involving maximizing, is also a legitimate one.

      In the domains of physics, problems that are posed using this method may refer to the technique as energy minimization. When doing so, they may talk of the value of the function f as reflecting the energy of the system that is being represented. When it comes to machine learning, it is always required to continually assess the quality of a data model by making use of a cost function. In this function, a minimum suggests a set of possible ideal parameters with an optimal (lowest) error.

      Typically, A is some subset of the Euclidean space ℝn, often governed by a predetermined list of restrictions, requirements of equality or inequality that must be met by the members of group A.

      The domain A of f is also referred to as the choice set or the search space, while the components of A are often referred to as candidate solutions or plausible solutions.

      Depending on the context, the function denoted by the letter f may be referred to as an objective function, a loss function or cost function (functions that minimize losses or costs), a utility function or fitness function (functions that maximize benefits), or, in some contexts, an energy function or energy functional. An optimum solution is referred to as a viable solution that optimizes (or maximizes, depending on whether or not that is the aim) the objective function.

      Conventional optimization issues in mathematics are often phrased in terms of minimization.

      A local minimum x* is defined as an element for which there exists some δ > 0 such that

      {\displaystyle \forall \mathbf {x} \in A\;{\text{where}}\;\left\Vert \mathbf {x} -\mathbf {x} ^{\ast }\right\Vert \leq \delta ,\,}

      the expression f(x*) ≤ f(x) holds; That is to say, on some portion of the area around x*, every value of the function is either higher than or equal to the value at that element. Local maxima are defined similarly.

      A global minimum is at least as excellent as any element that is even remotely possible, as contrast to a local minimum, which is just at least as good as any adjacent components. In a minimization issue, it is possible for there to be more than one local minimum, unless the objective function being minimized is convex. In a problem that is convex, if there is a local minimum that is interior (meaning that it is not on the edge of the set of feasible elements), then it is also the global minimum; however, in a problem that is nonconvex, there may be more than one local minimum, and not all of those local minima need to be global minima.

      Many of the proposed algorithms for solving nonconvex problems, as well as the vast majority of the solvers that are available for purchase, are incapable of differentiating between locally optimal solutions and globally optimal solutions, and as a result, they will only consider the former to be true answers to the initial problem. This is true even when the algorithms are designed to solve nonconvex problems. Global optimization is the area of applied mathematics and numerical analysis that focuses on the creation of deterministic algorithms that are able to guarantee convergence to the actual optimal solution of a nonconvex problem within a finite amount of time. This area of study is related to the study of convex optimization.

      Notation that is unique is often used when expressing optimization challenges. Here are several instances:

      Take into account the note that follows::

      {\displaystyle \min _{x\in \mathbb {R} }\;\left(x^{2}+1\right)}

      This denotes the minimum value of the objective function x² + 1, when choosing x from the set of real numbers ℝ.

      In this scenario, one is the absolute lowest possible value, occuring with x equal to zero.

      In the same vein, the notation

      \max _{x\in \mathbb {R} }\;2x

      requests the value that gives the highest benefit from the objective function 2x, where x may be any real integer. Since the objective function in this scenario is unbounded, there is no such thing as a maximum value; hence, the correct response is infinity or undefined..

      Take into account the note that follows::

      {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,

      or equivalently

      {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].

      This represents the value (or values) of the argument x in the interval (−∞,−1] that minimizes (or minimises) the objective function x² + 1 (the actual minimum value of that function is not what the problem asks for).

      This being the case, The solution is that x equals -1, as it is impossible for x to equal zero, that is, It is not part of the set of things that are viable.

      Similarly, {\displaystyle {\underset {x\in [-5,5],\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos y,}

      or equivalently

      {\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos y,\;{\text{subject to:}}\;x\in [-5,5],\;y\in \mathbb {R} ,}

      stands for the symbol x, y} pair (or pairs) that maximizes (or maximize) the value of the objective function x cos y, in addition to the restriction that x must fall within the range [5,5] (once more),, It makes no difference what the actual highest value of the expression is).

      This being the case, The pairings satisfying the conditions 5 constitute the solutions, 2kπ} and {−5, (2k + 1)π}, where k may take on any value among the integers.

      The operators arg min and arg max, which may alternatively be written as argmin and argmax, respectively, stand for the argument of the minimum and the argument of the maximum, respectively.

      Formulae based on calculus were discovered by Fermat and Lagrange, and iterative procedures were suggested by Newton and Gauss as a means of progressing toward optimality by these two mathematicians.

      Although the most of the theory had been presented by Leonid Kantorovich in 1939, George B. Dantzig is the one who is credited with coming up with the name linear programming to describe particular optimization scenarios. (The term programming does not apply to computer programming in this context; rather, it originates from the usage of the term program by the United States military to refer to planned training and logistical timetables, which were the challenges Dantzig examined at the time.) 1947 was the year when Dantzig released the Simplex algorithm, and John von Neumann was the one who invented the notion of duality in the same year.

      The following are some more noteworthy researchers that have contributed to mathematical optimization::

      Richard Bellman

      Dimitri Bertsekas

      Michel Bierlaire

      Roger Fletcher

      It was

      Enjoying the preview?
      Page 1 of 1