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1460 results in Control systems and optimization

Page 38 of 73

  • 4 - Analysis, Design, Optimization and Modeling

  • from PART II - USING OPTIMIZATION–THE ROAD MAP
  • Achille Messac, Mississippi State University
  • Book:
    Optimization in Practice with MATLAB®
    Published online:
    28 May 2018
    Print publication:
    19 March 2015, pp 82-92

  • Summary

    Overview

    In this chapter, we introduce the important activities of analysis, design, optimization, and modeling. While we are all generally familiar with these terms, it is important to understand them in the context of how they relate to our optimization activities. We also develop an important understanding of how modeling system behavior is a distinct activity from modeling an optimization problem. These involve two distinct lines of expertise. In most of your courses (e.g., structures, dynamics and finance) you focused on the former. In this book, we focus on the latter.

    Analysis, Design and Optimization

    Analysis and optimization are two activities integral to the process of design (Refs. [1, 2, 3, 4, 5]). In this chapter, as well as in the remainder of the book, we will primarily focus on these two activities in the context of engineering or systems design. However, the mathematical concepts, approaches, algorithms, and software tools that you will learn in this book could be applicable to diverse fields beyond engineering (e.g., optimization of market portfolios) [6]. Although, from a technical perspective, analysis and optimization can be considered to be steps within the process of design, they can also be performed as stand-alone activities toward other end goals within the scope of academic research and industrial R&D. In this section, you will learn the definition of these activities and how they are related to each other. At the same time, you will have the opportunity to understand and appreciate the roles and responsibilities of the users, researchers, or engineers who execute these activities, whether individually or as a part of a team.

    In order to help you better understand the practical essence of these activities, the following simple design example will be used throughout this section. You are designing a table that can carry as much weight (of objects placed on it) as possible, while fitting within a particular corner of your room. You are allowed a limited budget to construct the table.

  • 14 - Discrete Optimization

  • from PART V - MORE ADVANCED TOPICS IN OPTIMIZATION
  • Achille Messac, Mississippi State University
  • Book:
    Optimization in Practice with MATLAB®
    Published online:
    28 May 2018
    Print publication:
    19 March 2015, pp 335-354

  • Summary

    Overview

    In most previous chapters, continuous optimization problems were considered where the design variables were assumed to be continuous; that is, design variables assume real values within given ranges. In many practical engineering problems, the acceptable values of the design variables do not form a continuous set. These problems are referred to as discrete optimization problems. For example, the number of rivets required in a riveted joint has to be an integer (such as 1, 2, 3). Another example is when the feasible region of the design variable is a set of given discrete numbers, such as {6.25, 6.95, 7.65}, which may be the available standardized sizes of nuts. The basics of discrete optimization were introduced in Chapter 9, where some pertinent elementary methods were presented. This chapter introduces more advanced approaches. The reader is advised to first review Chapter 9 as preparation for the current chapter.

    This chapter is organized as follows. The next section (Sec. 14.2) provides the problem classes, examples, and definition (along with the notion of computational complexity of the solution algorithms). Section 14.3 discusses the basics of some popular techniques used to solve integer programming problems, with examples. The methods studied will be: the exhaustive search method (Sec. 14.3.1), the graphical method (Sec. 14.3.2), the relaxation method (Sec. 14.3.3), the branch and bound method (Sec. 14.3.4), and the cutting plane method (Sec. 14.3.5). Popular current software options (Sec. 14.3.7) are also discussed. The chapter concludes with a summary in Sec. 14.4.

    Problem Classes, Examples and Definition

    This section presents discrete optimization problem classes, problem examples, and problem definition. Computational complexity of the solution algorithms is also briefly addressed in connection with the discrete optimization problem definition.

  • 19 - Evolutionary Algorithms

  • from PART V - MORE ADVANCED TOPICS IN OPTIMIZATION
  • Achille Messac, Mississippi State University
  • Book:
    Optimization in Practice with MATLAB®
    Published online:
    28 May 2018
    Print publication:
    19 March 2015, pp 445-460

  • Summary

    Overview

    This book has presented various algorithms and applications where the optimizer was primarily gradient-based (i.e., the search direction is governed by gradient and/or Hessian information). This chapter introduces an entirely different class of optimization algorithms called the evolutionary algorithms (EA). Evolutionary algorithms imitate natural selection processes to develop powerful computational algorithms to select optimal solutions. Genetic algorithms (GA), simulated annealing (SA), ant colony optimization (ACO), particle swarm optimization (PSO), and tabu search (TS) are some of the popular techniques that fall under the umbrella of evolutionary algorithms.

    The motivation for using biologically-inspired computational approaches stems from two key observation. First, the mathematical optimization algorithms in solving complex problems in engineering, computing, and other fields suffer strong limitations. The common challenges in these areas revolve around the lack of mathematical models that define the physical phenomena, discontinuous functions, and high nonlinearity. Second, many complex problems encountered in engineering already exist in nature in some relevant form. Optimization is inherent in nature, such as in the process of adaptation performed by biological organisms in order to survive. Engineers and scientists continue to explore the various efficient problem-solving techniques employed by nature to optimize natural systems.

    The relative advantages and limitations of EAs vs. traditional optimization methods are as follows:

  • Traditional algorithms typically generate a single candidate optimum at each iteration that progresses toward the optimal solution. Evolutionary algorithms generate a population of points at each iteration. The best point in the population approaches an optimal solution.

  • Traditional algorithms calculate the candidate optimal point at the next iteration by a deterministic computation. EAs usually select the next population by a combination of operations that use random number generators.

  • Traditional algorithms require gradient and/or Hessian information to proceed, while EAs usually require only function values. As a result, EAs can solve a variety of optimization problems in which the objective function is not smooth and potentially discontinuous.[…]

  • 6 - Multiobjective Optimization

  • from PART III - USING OPTIMIZATION—PRACTICAL ESSENTIALS
  • Achille Messac, Mississippi State University
  • Book:
    Optimization in Practice with MATLAB®
    Published online:
    28 May 2018
    Print publication:
    19 March 2015, pp 123-157

  • Summary

    Overview

    In Part I of this book, we reviewed preparatory knowledge needed to start learning optimization: Matlab, and elementary mathematics. In Part II, we were exposed to the world of optimization and its potentially powerful role in our lives as engineers or professionals in quantitative fields. We then delved into the specific activities of analysis, design, and optimization, their links and distinctions. In the previous chapter, we then began addressing the fundamental aspects of computational optimization. These included single objective optimization, and the different approaches to optimization (analytical, numerical, experimental, and graphical). This was followed by a discussion of software options.

    In this chapter, we study one of the most important aspects of optimization in practice, the notion of multiobjective optimization. Stated simply, Multiobjective optimization is the art and science of formulating how to optimize a set of competing objectives, which is almost always the case in practice. A detailed presentation of the pertinent methods is provided in such a way as to allow you to be readily productive and effective in practical design.

    The Multiobjective Problem Definition

    The identification of the right design objectives plays a crucial role in the design of any system. More often than not, in real-life design, you will find that your optimization problem contains more than one design objective. For example, wouldn't it be nice if your car dealer would tell you that the car you like happens to feature more miles per gallon and also costs less than a competitor's car? As an engineer, you would think twice before making a decision based on his interesting comments. If he were right, the other car company would probably not stay in business for very long. The point here is that while designing any product or system, you will almost always have to consider several competing design objectives. As a car designer, you would like the car to provide the most miles per gallon possible, while taking care that the car does not cost a million dollars.

  • Preface

  • Achille Messac, Mississippi State University
  • Book:
    Optimization in Practice with MATLAB®
    Published online:
    28 May 2018
    Print publication:
    19 March 2015, pp xxv-xxx

  • Summary

    Contacting the Author Regarding this Book

    I am delighted that you are using this book in your study of, or involvement with, optimization. I would very much welcome your comments, particularly regarding this first edition, and any suggestions that you might have for the next edition. For your comments regarding this book, I will be happy to receive your direct email at [email protected].

    Book Website

    The website www.cambridge.org/Messac will be maintained for this book. Information for instructors and students will be separately provided. Software for various problems will be provided in this website. I expect it to be a dynamic website, where the information available will evolve over time to be responsive to readers’ requests and feedback.

    Book Organization

    This book is intended to be used by undergraduate and graduate students in the classroom, or by industry practitioners learning independently. Its organization suits these objectives. Following are messages specifically tailored for students, for industry practitioners and for instructors. The book has five parts:

  • Part I Helpful Preliminaries

  • Part II Using Optimization—The Road Map

  • Part III Using Optimization—Practical Essentials

  • Part IV Going Deeper: Inside the Codes and Theoretical Aspects

  • Part V More Advanced Topics in Optimization

    • Part I has two chapters that present prerequisite material explaining how to use Matlab and some useful mathematical information. Most of this book assumes knowledge of undergraduate calculus and elementary linear algebra. The second chapter provides a brief review of the math needed. In Part II, three chapters introduce the world of optimization in the form of a road map. This part provides the basics of what should be known about optimization before attempting to use it. In Part III, there are five chapters that teach the basic use of optimization. In fact, learning the material up to Part III provides the practitioner with sufficient information for solving practical problems. In doing so, student users will not be experts on how optimization works under the hood, so to speak, but will have the ability to use optimization in general practical contexts.

  • 18 - Physical Programming for Multiobjective Optimization

  • from PART V - MORE ADVANCED TOPICS IN OPTIMIZATION
  • Achille Messac, Mississippi State University
  • Book:
    Optimization in Practice with MATLAB®
    Published online:
    28 May 2018
    Print publication:
    19 March 2015, pp 429-444

  • Summary

    Overview

    Engineering design problems are multiobjective in nature. These problems usually optimize two or more conflicting objectives – simultaneously. An approach to multiobjective problem formulation combines the multiple objectives into a single objective function, also known as the Aggregate Objective Function (AOF). This AOF is solved to obtain one Pareto solution. One of several challenges in the area of multiobjective optimization is to judiciously construct an AOF that satisfactorily models the designer's preferences. This chapter provides a concise presentation of the Physical Programming method, which defines a framework to effectively incorporate the designer's preferences into the AOF (Ref. [1]).

    Several methods to solve multiobjective optimization problems have been discussed in Chapter 6, such as the weighted sum method, compromise programming, and goal programming. These weight-based approaches require the designer to specify numerical weights in defining the AOF. This process can be ambiguous. For example, consider the following: (1) How does the designer specify weights in weight-based approaches? (2) Do the weights reflect the designer's preferences accurately? If the designer chooses to increase the importance of a particular objective, by how much should he/she increase the weight? Is 25% adequate? Or is 200% adequate? (3) Does the AOF denote a true mathematical representation of the designer's preferences?

    The above questions begin to explain that the problem of determining “good weights” can be difficult and dubious. Because of this ambiguity, the weight selection process is often a computational bottleneck in large scale multiobjective design optimization problems. The above discussion paves the way for a multiobjective problem formulation framework that alleviates these ambiguities: Physical Programming (PP) developed by Messac [2].

    Physical Programming systematically develops an AOF that effectively reflects the designer's wishes. This approach eliminates the need for iterative weight setting, which alleviates the above discussed ambiguities. Instead of choosing weights, the designer chooses ranges of desirability for each objective. The PP method formulates the AOF from these ranges of desirability, while yielding interesting and useful properties for the AOF.

  • 3 - Optimal power flow

  • Joshua Adam Taylor, University of Toronto
  • Book:
    Convex Optimization of Power Systems
    Published online:
    28 May 2018
    Print publication:
    12 February 2015, pp 43-80

  • Summary

    In the late 1880s, Thomas Edison and George Westinghouse fought the so-called War of the Currents to decide whether the incumbent direct current or Nicola Tesla's alternating current technology would become the standard for future power systems. Endnotes [1, 2] provide excellent historical accounts. The winning argument was that it was much easier using the technology of the time, transformers, to change voltage levels in AC power systems, enabling efficient high-voltage transmission of power and lower voltage generation and end usage (see (2.6) in Section 2.5.1). Beyond pride, part of Edison's opposition to AC transmission was rooted in the higher level of mathematics necessary to understand it.

    Today, power electronics have enabled direct current to make a comeback in certain applications like long-distance transmission and microgrids, cf. Section 3.5.1. Some even say that we are now constrained by the mathematical model of AC power flow, which while simple to write down is a quagmire for analysis and computation. Here we tackle this issue head on in one of its purest forms, optimal power flow.

    In words, optimal power flow is the problem of minimizing some function of voltage, current, and power, subject to the resulting flow being able to feasibly traverse a transmission or distribution system. Since its introduction by Carpentier [3], virtually every algorithm for continuous optimization has been applied, cf. endnotes [4–12] and the surveys [13–15]. Optimal power flow had similar but separate beginnings in the Russian academic literature [16]. System operators solve optimal power flow routines to do long-term planning, days- to hour-ahead scheduling, real-time dispatch, and pricing (to name a few), making it one of the most frequently employed optimization routines in power systems. As will be seen over the course of this book, many other power system optimizations are essentially optimal power flow models with additional layers of detail.

  • 5 - Infrastructure planning

  • Joshua Adam Taylor, University of Toronto
  • Book:
    Convex Optimization of Power Systems
    Published online:
    28 May 2018
    Print publication:
    12 February 2015, pp 112-131

  • Summary

    This chapter zooms out to consider the design of power systems. For example, if a city's power consumption has grown in support of new industry or increased population, should it become the home of a new generator or the destination of a new transmission line? What if there are 100 such choices, all within the same interconnected power system? Additionally, each generation project will take five years to complete and each transmission project ten, over which the energy consumption landscape may change significantly. Finally, each element of the design must respect environmental restrictions, harmonize with a potentially poorly designed legacy system, and be separately financed. It's no wonder that the evolution of power systems looks as much like the product of organic synthesis as deliberate design. It is hence somewhat more reasonable to plan power systems and then hope that plans stay consistent with reality to yield efficient and reliable infrastructures.

    Even in the absence of the factors listed above, power system planning remains a source of intractable problems because it almost always entails the use of integer variables. Unfortunately, any plan that builds a one-micron-thick transmission line or a two-watt nuclear power plant is not very useful. This makes many power system planning problems NP-hard, and thus very hard to solve. Fortunately, although these problems cannot be solved in polynomial time, powerful mixed-integer programming heuristics like cutting planes and branch-and-bound make moderate-sized problems tractable.

    This chapter ignores many of the above real-world details of power system planning and focuses on its underlying mathematical skeleton. This approach is to respect discrete constraints and seek formulations that have convex continuous relaxations, as discussed in Section 2.2.4. More precisely, if the problem we'd like to solve is

    minimize f(x, y)

    subject to gi(x, y) ≤ 0

    yi ∈ ℤ,

    then we want f(x, y) and each gi(x, y) to be convex.

  • 2 - Background

  • Joshua Adam Taylor, University of Toronto
  • Book:
    Convex Optimization of Power Systems
    Published online:
    28 May 2018
    Print publication:
    12 February 2015, pp 5-42

  • Summary

    This chapter summarizes the basic technical concepts used throughout this book. As stated in the Introduction, this book focuses on modeling, so most algorithmic aspects are left “under the hood.” Because this book is intended to appeal to anyone familiar with power systems or optimization, background material on both topics is covered, albeit at the minimum depth necessary to access the later material.

    Convexity and computational complexity

    We begin with a few core concepts. A point x0 ∈ X is a global minimum of the function f(x) over the set X ⊆ ℝn if f(x0) ≤ f(x) for all x ∈ X. If f(x) is continuous and X is compact, which is to say closed and bounded, such a point is guaranteed to exist. x0 is a local minimum of f(x) if there exists an ε > 0 for which f(x0) ≤ f(x) for all x ∈ X satisfying ∥x − x0∥ ≤ ε. All minima of a convex function achieve the same function value and are therefore global. In general, a function may have multiple local and global minima.

    To find a global minimum of a convex function, choose a descending algorithm, let it run free, and in a perfect world it will eventually end up there. (In the real world, large problem sizes or bad numerical conditioning can derail any algorithm.) The intuitive simplicity of convexity translates to a genuine computational advantage, which is evinced by the powerful algorithms that exist for convex optimization and the extreme difficulty of nonconvex optimization. This section gives some basic characterizations of convexity and describes the varieties of convex optimization problems encountered in this book. For more comprehensive coverage, the reader is referred to endnotes [1–5], and to endnotes [6, 7] for theoretical treatments of convex functions and sets.

    A function f is convex if, for any two points in its domain, x and y,

    f(αx + (1 − α)y) ≥ αf(x) + (1 − α)f(y) for all α ∈ [0, 1].

Page 38 of 73