Mathematics
Optimization Problems
Optimization problems involve finding the best solution from a set of feasible options. In mathematics, these problems often revolve around maximizing or minimizing a certain quantity, subject to given constraints. They are commonly encountered in various fields such as engineering, economics, and operations research, and are solved using techniques from calculus, linear algebra, and optimization theory.
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eBook - PDFMaxima and Minima with Applications
Practical Optimization and Duality
- Wilfred Kaplan(Author)
- 2011(Publication Date)
- Wiley-Interscience(Publisher)
3 Optimization Optimization is the goal of all practical design projects, although few achieve such an ultimate state of perfection. Commonly a design is finalized signifi-cantly short of its optimum as a result of a complex set of compromises made for economic, technical, or social reasons. The decision-making process is usually a mixture of analysis, experiments, and intuition. When the variables in a problem become numerous, experiments are expensive, intuition breaks down, and a systematic approach is necessary. Using the modern theory of opti-mization, one can often model the relationships between the variables mathematically and then, taking advantage of rapid computers, seek the optimal solution of the model problem numerically. Many practical problems are naturally formulated as problems of minimiza-tion or maximization of (1) real functions of several real variables, or (2) of integrals depending on unknown functions. In the first case, one has a finite-dimensional problem, as in the previous chapters. In the second case one has a problem of the calculus of variations, an infinite-dimensional problem; however, methods of numerical analysis, especially those using finite elements, convert the problem into a finite-dimensional one. These finite-dimensional problems all fall in the area of mathematical programming. This area also includes maximum and minimum problems arising in operations research, notably those of linear programming. In this chapter we consider some important problems of mathematical pro-gramming, treating them in order of increasing difficulty: quadratic, linear, convex, and general nonlinear problems. 3.1 CONVEXITY We here review the ideas about convex sets and convex or concave functions presented in Chapters 1 and 2 and introduce some new properties. 135 136 Optimization A set E in ΊΖ η is said to be convex if, for every pair of points in E, the line segment joining them is contained in E.
eBook - PDF- Xin-She Yang(Author)
- 2008(Publication Date)
- WSPC(Publisher)
Part IV Mathematical Programming This page intentionally left blank This page intentionally left blank Chapter 13 Mathematical Optimization Optimization is everywhere, from business transactions to engineering de-sign, from planning your holiday to your daily routine. Business organi-sations have to maximize their profit and minimize the cost. Engineering design has to maximize the performance of the designed product while of course minimizing the cost at the same time. Even when we plan holidays we want to maximize the enjoyment and minimize the cost. Therefore, studies of optimization are of both scientific interest and practical implica-tions and subsequently the methodology will have many applications. 13.1 Optimization Whatever the real world problem is, it is usually possible to formulate the optimization problem in a generic form. All Optimization Problems with explicit objectives can in general be expressed as nonlinearly constrained Optimization Problems in the following generic form maximize/minimize x ∈ℜ n f ( x ) , x = ( x 1 ,x 2 ,...,x n ) T ∈ ℜ n , subject to φ j ( x ) = 0 , ( j = 1 , 2 ,...,M ) , ψ k ( x ) ≥ 0 , ( k = 1 ,...,N ) , (13.1) where f ( x ), φ i ( x ) and ψ j ( x ) are scalar functions of the real column vector x . Here the components x i of x = ( x 1 ,...,x n ) T are called design or decision variables, and they can be either continuous, or discrete or mixed of these two. The vector x is often called the decision vector which varies in an n -dimensional space ℜ n . The function f ( x ) is called the objective function or cost function. In addition, φ i ( x ) are constraints in terms of M equalities, 143 144 Introduction to Computational Mathematics and ψ j ( x ) are constraints written as N inequalities. So there are M + N constraints in total. The optimization problem formulated here is a nonlinear constrained problem.
eBook - PDF- Mahmoud M. El-Halwagi(Author)
- 2006(Publication Date)
- Academic Press(Publisher)
2005; Grossmann and Biegler 2005; Floudas et al., 2004). This chapter presents an overview of using mathematical techniques to formulate Optimization Problems. Additionally, the use of optimization software to solve and analyze optimization programs will be presented. 11.1 MATHEMATICAL PROGRAMMING Over the past few decades, significant progress has been made in the field of mathematical programming which deals with the formulation, solution, and analysis of Optimization Problems or mathematical programs. The analysis and solution of an optimization problem may involve graphical, algebraic, or computer-aided tools. As such the word ‘‘programming’’ does not necessarily entail the use of computers or computer coding although in solving complex Optimization Problems, the use of computer-aided tools is becoming a common practice. A mathematical program (or a mathematical programming model or an optimization problem) can be represented by: min ðor maxÞ : fðx 1 ; x 2 . . . ; x N Þ (P11.1) Subject to: Inequality constraints g 1 ðx 1 ; x 2 ; . . . ; x N Þ 0 g 2 ðx 1 ; x 2 ; . . . ; x N Þ 0 . . . g m ðx 1 ; x 2 ; . . . x N Þ 0 Equality constraints h 1 ðx 1 ; x 2 ; . . . ; x N Þ ¼ 0 h 2 ðx 1 ; x 2 ; . . . ; x N Þ ¼ 0 . . . h E ðx 1 ; x 2 ; . . . x N Þ ¼ 0 E N The variables x 1 , x 2 , . . ., x n are referred to as decision or optimiza- tion variables. In a vector notation, a mathematical program can be 286 CHAPTER 11 OVERVIEW OF OPTIMIZATION expressed as: min ðor maxÞ fðxÞ where x T ¼ ½x 1 ; x 2 ; . . . x N (P11.2a) Subject to: gðxÞ 0 where g T ¼ ½g 1 ; g 2 ; . . . g m hðxÞ ¼ 0 where h T ¼ ½h 1 ; h 2 ; . . . h E The vector x is referred to as the vector of optimization vari- ables. It is also possible to describe the constraints only as inequality constraints, i.e., min ðor maxÞ fðxÞ where x T ¼ ½x 1 ; x 2 ; .
- Alejandro Garces Ruiz(Author)
- 2021(Publication Date)
- Wiley-IEEE Press(Publisher)
17 Part I Mathematical programming 19 2 A brief introduction to mathematical optimization Learning outcomes By the end of this chapter, the student will be able to: ● Establish first-order conditions for locally optimal solutions. ● Solve unconstrained Optimization Problems, using the gradient method, implemented in Python. ● Solve equality-constrained Optimization Problems, using Newton’s method implemented in Python. 2.1 About sets and functions Sets and functions are familiar concepts in mathematics. A set is a well-defined collection of distinct objects, considered an object in its own right. A function is a map that takes objects from one set (i.e., input or domain) and returns an object in another set (i.e., output or image). An optimization problem consists of finding the best object in the output set and its corresponding input, as shown schematically in Figure 2.1. The input set Ω is called the set of feasible solutions, and the best object corresponds to a minimum or a maximum, according to an objective function ? ∶ Ω ⊆ ℝ ? → ℝ . Solving an optimization problem implies not only to find the value of the objective function (e.g., ? min or ? max ) but also the value ? , at the input set Ω (e.g., ? min , ? max ). These values are represented as follows: min ? ∈Ω ? ( ? ) = ? min , max ? ∈Ω ? ( ? ) = ? max (2.1) argmin ? ∈Ω ? ( ? ) = ? min , argmax ? ∈Ω ? ( ? ) = ? max (2.2) Mathematical Programming for Power Systems Operation: From Theory to Applications in Python . First Edition. Alejandro Garcés. © 2022 by The Institute of Electrical and Electronics Engineers, Inc. Published 2022 by John Wiley & Sons, Inc. 20 2 A brief introduction to mathematical optimization min max min max Ω , input set, domain or set offeasible solutions output set or image , Figure 2.1 Representation of the sets related to a general optimization problem. Notice that ? min and ? max are numbers whereas ? min and ? max are vectors.
eBook - ePub- Walter J. Meyer(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
FOUROPTIMIZATION
1 CLASSICAL OPTIMIZATION
Abstract We begin with the concepts of feasible set and objective function and then proceed to review calculus techniques for optimization. Search methods make a brief appearance in the exercises. Emphasis is placed on the role of the boundary points as candidates for optimization.Prerequisites Calculus of one and several variables.Mathematical models are often used to help us make decisions. When we use a mathematical model to select the best alternative out of a large number of possibilities, we call this optimization .Optimization is surely one of the most basic human activities. When a child is asked to choose 1 candy out of a handful of 10, the child faces a problem in optimization. Should the choice be the one with the largest volume, the heaviest, the one with the most chocolate? Having chosen a criterion, the child then needs to find the candy that measures up the best with respect to that criterion.This example, simple though it is, illustrates two features which are also found in more complicated mathematical models: the feasible set and the objective function. DefinitionThe feasible set is the set of all the possibilities from which we are allowed to choose.In our example, the 10 candies form the feasible set. DefinitionThe objective function is a function which assigns to each member of the feasible set a number which measures the desirability of that choice.For example, suppose the child decided on the candy with the largest volume. Then volume would be the objective function. Each candy is assigned a number equal to its volume. The candy with the largest number wins. A child would not normally measure the volumes precisely—an eyeball estimate would be more likely—but in mathematical models we usually have a formula for the objective function.Optimization sometimes involves minimization instead of maximization. For example, the child might wish to minimize the amount of nuts.
eBook - PDFSpreadsheet Modeling & Decision Analysis
A Practical Introduction to Business Analytics
- Cliff Ragsdale(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Expressing Optimization Problems Mathematically 19 truck can carry from one warehouse to the stores on its route. In the fourth example, laws determine the minimum and maximum amounts that can be withdrawn from retirement accounts without incurring a penalty. Many other constraints can also be identified for these examples. Indeed, it is not unusual for real-world Optimization Problems to have hundreds or thousands of constraints. A final common element in each of the examples is the existence of some goal or objective that the decision maker considers when deciding which course of action is best. In the first example, the production manager can decide to produce several different product mixes given the available resources, but the manager will probably choose the mix of products that maximizes profits. In the second example, a large number of possible drilling patterns can be used, but the ideal pattern will probably involve moving the drill bit the shortest total distance. In the third example, there are numerous ways merchandise can be shipped from the warehouses to supply the stores, but the company will probably want to identify the routing that minimizes the total transportation cost. Finally, in the fourth example, individuals can withdraw money from their retirement accounts in many ways without violating tax laws, but they probably want to find the method that minimizes their tax liability. 2.3 Expressing Optimization Problems Mathematically From the preceding discussion, we know that Optimization Problems involve three ele-ments: decisions, constraints, and an objective. If we intend to build a mathematical model of an optimization problem, we will need mathematical terms or symbols to represent each of these three elements. 2.3.1 DECISIONS The decisions in an optimization problem are often represented in a mathematical model by the symbols X 1 , X 2 , .
eBook - PDFSpreadsheet Modeling and Decision Analysis
A Practical Introduction to Business Analytics
- Cliff Ragsdale(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Businesses also have limited resources. A manufacturing organization employs a limited number of workers. A restaurant has a limited amount of space available for seating. Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 17 Applications of Mathematical Optimization Deciding how best to use the limited resources available to an individual or a business is a universal problem. In today’s competitive business environment, it is increasingly important to make sure that a company’s limited resources are used in the most efficient manner possible. Typically, this involves determining how to allocate the resources in such a way as to maximize profits or minimize costs. Mathematical programming (MP) is an area in business analytics that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual or a business. More generally, MP is used to identify the best or optimal values for the decision variables in a mathematical model. For this reason, MP is often referred to as optimization and it is the driving force behind prescriptive analytics. It has been said that every decision problem is an optimization problem. That is, whenever we face a decision problem, we generally try to make the best decision possible under the cir- cumstances. As a result, optimization plays a key role in all analytics modeling techniques.
- William Bober, Andrew Stevens(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
295 Chapter 10 Optimization 10.1 Introduction The objective of optimization is to maximize or minimize some function f (called the object function ). You are probably familiar with determining the maxima and min-ima of functions using techniques from differential calculus. However, the problem becomes more complicated when we place constraints on the allowable solutions to f . As an example, suppose there is an electronics company that manufactures several different types of circuit boards. Each circuit board must pass through several dif-ferent departments (such as drilling, pick-and-place, testing, etc.) before shipping. The time required for each circuit board to pass through the various departments is also known. There is a minimum production quantity per month that the company must produce. However, the company is capable of producing more than the mini-mum production requirement for each type of circuit board each month. The profit the company will make on each circuit board it produces is known. The problem is to determine the production amount of each type of circuit board per month that will result in the maximum profit. A similar type of problem may be one in which the object is to minimize the cost of producing a particular product. These types of Optimization Problems are discussed in greater detail in this chapter. In most Optimization Problems, the object function f will usually depend on several variables, , , , , 1 2 3 x x x x n … . These are called the control variables because their values can be selected. Optimization theory develops methods for selecting optimal values for the control variables , , , , 1 2 3 x x x x n … that either maximize or minimize the objective function f . In many cases, the choice of values for , , , , 1 2 3 x x x x n … is not entirely free but is subject to some constraints.
eBook - ePubOptimization with LINGO-18
Problems and Applications
- Neha Gupta, Irfan Ali(Authors)
- 2021(Publication Date)
- CRC Press(Publisher)
Chapter 5Linear Optimization Problems
5.1 IntroductionA special case of mathematical optimization is linear optimization or linear programming or in short LP. LP is used to optimize the linear function known as “Objective Function” (i.e., performance, cost, profit, time, return on investment, among others) subject to a set of linear equalities or inequalities known as “Constraints,” on the use of limited resources (i.e., space, energy, time, material, labour, among others). The intersection of these linear equalities or inequalities form a feasible region, which is a convex polytope. Here, “linear” refers to the linear relationship between the variables; that is, if demand increases two times, then the profit also increases proportionally, while the meaning of “programming” is to convert the verbal problem and given data into a mathematical model.George B Dantzing first developed the LP model during World War II to derive the solutions of US military and air force tactical and strategic problems. But, now the LP problems are widely used in the fields of human resources, finance, management, health care, education, military, agriculture, scheduling problems, transportation, and many more. Various software packages are available to solve complex LP models such as R, TORA, SAS, MATLAB, LINDO, and more. But the formulation of the LP model cannot be done by any of these softwares. For modelling, one has to practice and understand the problem clearly. The general structure and basic definitions of the LP model are discussed in the next section.
Generally, the LP model consists of three components, which are described in the following subsections.5.2 General Form of LP Model5.2.1 Components of LP Model- Decision Variables: Decision variables are the quantities that are to be determined Usually they are denoted by x1 ,x2 ,…,xn
- Ali R. Al-Roomi(Author)
- 2021(Publication Date)
- Wiley-IEEE Press(Publisher)
1 1 Fundamental Steps in Optimization Algorithms Aim One of the main barriers that make ORC very difficult to understand is its dependency on one or multiple optimization stages. This chapter gives a quick overview. It explains the meaning of optimiza-tion algorithms, the main types and subtypes, the design constraints, the differences between classical and meta-heuristic algorithms, etc. Although this chapter belongs to applied mathematics and data science, it is very essential to be added here before starting our journey into ORC. Failing in under-standing the core of this chapter will have direct impacts and multiple consequences on understanding the rest of the book. 1.1 Overview The term mathematical optimization , or just optimization , is frequently heard in mathemat-ics, computer science, engineering, and even in economic and management science. Also, it can be found in proceedings, journals, books, encyclopedias, websites, etc., under different sections and names, like soft computing , applied mathematics and optimization , evolutionary computation , numerical analysis , etc. From the basic of mathematics, suppose that there is a function ( f ) and it changes as the independent variable ( x ) changes, then f becomes the dependent variable of x and known shortly as f ( x ) . Based on the system requirements, or in another word the objective function , the best solution to such a problem is called the optimum (or optimal ) solution . This solution is located at a spec-ified value of the design variable “ x .” The optima could be either maxima or minima , and the tool used to find this point is called an optimization algorithm . Example 1.1 Consider the following arbitrary function: f ( x ) = 30 − x + sin ( x ) + 5 20 x + e 0.3 x ; 0 ⩽ x ⩽ 10 Design a MATLAB code to graphically represent this function when x moves from 0 to 10 with steps of Δ x = 0.01. Then, use the command fsolve to find the global minimum and its solution.
eBook - PDF- Atila Ertas(Author)
- 2018(Publication Date)
- Wiley(Publisher)
The appropriate optimization method can now be applied to the above objective function (with the given constraints) to define the volume region that includes the optimum-design point. Optimum-design problems may involve the following: 1) Equality constraints only 2) Inequality constraints only 3) Both equality and inequality constraints 4) No constraint. For case (4), which is called an unconstrained-optimization problem, there are no restrictions imposed on the design variables. In general, there are two types of Optimization Problems: 1) Single-criterion Optimization Problems. Problems in which the designer’s goal is to minimize or maximize one objective function. In this situation the optimum point is simply the mini-mum or maximum. 2) Multicriterion Optimization Problems. Problems in which the designer’s goal is to minimize or maximize more than one objective function simultaneously. In this situation, all the objective functions should be considered to find the optimum solution. 9.2 Mathematical Models and Optimization Methods The mathematical methods for optimum design can be divided into two categories: 1) Analytical methods. This method includes differentiation, variational methods, and the use of Lagrange multipliers. 2) Numerical methods. This method includes linear (simplex method) and nonlinear program-ming methods such as those given in Figure 9.2. 1 This chapter discuses some of the common analytical and numerical optimization methods used in design. Additional information on these methods can be found in references 1, 3, and 5 in the chapter bibliography. 9.2.1 The Differential Calculus Method The differential calculus method uses the first and second derivatives to find maximum and min-imum values of a given differentiable function U ( x ) .
eBook - PDFMathematica Navigator
Mathematics, Statistics and Graphics
- Heikki Ruskeepaa(Author)
- 2009(Publication Date)
- Academic Press(Publisher)
23.2 Linear Optimization 23.2.1 Linear Problems by Variables We can formulate a linear programming problem either by writing explicit expressions with variables or by giving only coefficient matrices and vectors. The corresponding commands are Minimize or Maximize and LinearProgramming , respectively. Linear optimization is a special case of global optimization considered in Section 23.1. ‡ Formulation with Variables Minimize[{f, cons}, vars] Give the global minimum of f subject to constraints cons Minimize[{f, cons}, vars, dom] Minimize over domain dom Maximize[{f, cons}, vars] Give the global maximum of f subject to constraints cons Maximize[{f, cons}, vars, dom] Maximize over domain dom By specifying the domain to be Integers , we can solve integer problems. Within the constraints we can also restrict individual variables to be integer valued by writing, for example, x oe Integers . Note that Minimize and Maximize give an exact solution if the problem is exact ~ that is, does not contain decimal numbers. If the problem contains a decimal number, then actually NMinimize and NMaximize are used. Here is an example: Maximize @8 x + y,2x + y § 2, x -y ¥ -1 ê 2, x + 2y § 2 < , 8 x, y > The problem can be visualized as follows: Show @ RegionPlot @ 2x + y § 2&&x -y ¥ -1 ê 2&&x + 2y § 2, 8 x, 0, 1 < , 8 y, 0, 0.9 < , AspectRatio Ø Automatic D , ContourPlot @ x + y, 8 x, 0, 1 < , 8 y, 0, 0.9 < , ContourShading Ø False, Contours Ø Range @ 1 ê 3, 5 ê 3, 1 ê 3 D , ContourStyle Ø Dashing @ Small DD , Graphics @8 Red, PointSize @ Medium D , Point @8 2 ê 3, 2 ê 3
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