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37588 results in Cambridge Textbooks

Page 1476 of 1880

  • Preface

  • Giuseppe C. Calafiore, Politecnico di Torino, Laurent El Ghaoui, University of California, Berkeley
  • Book:
    Optimization Models
    Published online:
    28 May 2018
    Print publication:
    31 October 2014, pp xi-xviii

  • Summary

    Optimization refers to a branch of applied mathematics concerned with the minimization or maximization of a certain function, possibly under constraints. The birth of the field can perhaps be traced back to an astronomy problem solved by the young Gauss. It matured later with advances in physics, notably mechanics, where natural phenomena were described as the result of the minimization of certain “energy” functions. Optimization has evolved towards the study and application of algorithms to solve mathematical problems on computers.

    Today, the field is at the intersection of many disciplines, ranging from statistics, to dynamical systems and control, complexity theory, and algorithms. It is applied to a widening array of contexts, including machine learning and information retrieval, engineering design, economics, finance, and management. With the advent of massive data sets, optimization is now viewed as a crucial component of the nascent field of data science.

    In the last two decades, there has been a renewed interest in the field of optimization and its applications. One of the most exciting developments involves a special kind of optimization, convex optimization. Convex models provide a reliable, practical platform on which to build the development of reliable problem-solving software. With the help of user-friendly software packages, modelers can now quickly develop extremely efficient code to solve a very rich library of convex problems.

  • 7 - Matrix algorithms

  • from I - Linear algebra models
  • Giuseppe C. Calafiore, Politecnico di Torino, Laurent El Ghaoui, University of California, Berkeley
  • Book:
    Optimization Models
    Published online:
    28 May 2018
    Print publication:
    31 October 2014, pp 199-220

  • Summary

    In this chapter we present a compact selection of numerical algorithms for performing basic matrix computations. Specifically, we describe the power iteration method for computing eigenvalues and eigenvectors of square matrices (along with some of its variants, and a version suitable for computing SVD factors); we discuss iterative algorithms for solving square systems of linear equations, and we detail the construction of the QR factorization for rectangular matrices.

    7.1 Computing eigenvalues and eigenvectors

    7.1.1 The power iteration method

    In this section we outline a technique for computing eigenvalues and eigenvectors of a diagonalizable matrix. The power iteration (PI) method is perhaps the simplest technique for computing one eigenvalue/eigenvector pair for a matrix. It has rather slow convergence and it is subject to some limitations. However, we present it here since it forms the building block of many other more refined algorithms for eigenvalue computation, such as the Hessenberg QR algorithm, and also because interest in the PI method has been recently revived by applications to very large-scale matrices, such as the ones arising in web-related problems (e.g., Google PageRank). Many other techniques exist for computing eigenvalues and eigenvectors, some of them tailored for matrices with special structure, such as sparse, banded, or symmetric. Such algorithms are described in standard texts on numerical linear algebra.

  • 1 - Introduction

  • Giuseppe C. Calafiore, Politecnico di Torino, Laurent El Ghaoui, University of California, Berkeley
  • Book:
    Optimization Models
    Published online:
    28 May 2018
    Print publication:
    31 October 2014, pp 1-18

  • Summary

    Optimization is a technology that can be used to devise effective decisions or predictions in a variety of contexts, ranging from production planning to engineering design and finance, to mention just a few. In simplified terms, the process for reaching the decision starts with a phase of construction of a suitable mathematical model for a concrete problem, followed by a phase where the model is solved by means of suitable numerical algorithms. An optimization model typically requires the specification of a quantitative objective criterion of goodness for our decision, which we wish to maximize (or, alternatively, a criterion of cost, which we wish to minimize), as well as the specification of constraints, representing the physical limits of our decision actions, budgets on resources, design requirements that need be met, etc. An optimal design is one which gives the best possible objective value, while satisfying all problem constraints.

    In this chapter, we provide an overview of the main concepts and building blocks of an optimization problem, along with a brief historical perspective of the field. Many concepts in this chapter are introduced without formal definition; more rigorous formalizations are provided in the subsequent chapters.

    1.1 Motivating examples

    We next describe a few simple but practical examples where optimization problems arise naturally. Many other more sophisticated examples and applications will be discussed throughout the book.

  • 15 - Control problems

  • from III - Applications
  • Giuseppe C. Calafiore, Politecnico di Torino, Laurent El Ghaoui, University of California, Berkeley
  • Book:
    Optimization Models
    Published online:
    28 May 2018
    Print publication:
    31 October 2014, pp 567-590

  • Summary

    Dynamical systems are physical systems evolving in time. Typically, these systems are modeled mathematically by a system of ordinary differential equations, in which command and disturbance signals act as inputs that, together with pre-existing initial conditions, generate the evolution in time of internal variables (the states) and of output signals. These kinds of model are ubiquitous in engineering, and are used to describe, for instance, the behavior of an airplane, the functioning of a combustion engine, the dynamics of a robot manipulator, or the trajectory of a missile.

    Broadly speaking, the problem of control of a dynamical system amounts to determining suitable input signals so as to make the system behave in a desired way, e.g., to follow a desired output trajectory, to be resilient to disturbances, etc. Even an elementary treatment of control of dynamical systems would require an entire text-book. Here, we simply focus on very few specific aspects related to a restricted class of dynamical systems, namely finite-dimensional, linear, and time-invariant systems.

    We start our discussion by introducing continuous-time models and their discrete-time counterparts. For discrete-time models, we highlight the connections between the input-output behavior over a finite horizon, and static linear maps described by systems of linear equations. We shall show how certain optimization problems arise naturally in this context, and discuss their interpretation in a control setting.

  • 2 - Vectors and functions

  • from I - Linear algebra models
  • Giuseppe C. Calafiore, Politecnico di Torino, Laurent El Ghaoui, University of California, Berkeley
  • Book:
    Optimization Models
    Published online:
    28 May 2018
    Print publication:
    31 October 2014, pp 21-54

  • Summary

    Le contraire du simple n'est pas

    le complexe, mais le faux.

    André Comte-Sponville

    A vector is a collection of numbers, arranged in a column or a row, which can be thought of as the coordinates of a point in n-dimensional space. Equipping vectors with sum and scalar multiplication allows us to define notions such as independence, span, subspaces, and dimension. Further, the scalar product introduces a notion of the angle between two vectors, and induces the concept of length, or norm. Via the scalar product, we can also view a vector as a linear function. We can compute the projection of a vector onto a line defined by another vector, onto a plane, or more generally onto a subspace. Projections can be viewed as a first elementary optimization problem (finding the point in a given set at minimum distance from a given point), and they constitute a basic ingredient in many processing and visualization techniques for high-dimensional data.

    2.1 Vector basics

    2.1.1 Vectors as collections of numbers

    Vectors are a way to represent and manipulate a single collection of numbers. A vector x can thus be defined as a collection of elements x1, x2, …, xn, arranged in a column or in a row. We usually write vectors in column format:

    Element xi is said to be the i-th component (or the i-th element, or entry) of vector x, and the number n of components is usually referred to as the dimension of x.

  • 9 - Exporting Through Licensing and Franchising Arrangements

  • Justin Malbon, Monash University, Victoria, Bernard Bishop, Griffith University, Queensland
  • Book:
    Australian Export
    Published online:
    06 August 2018
    Print publication:
    30 October 2014, pp 249-281

  • Summary

    Key Concepts

  • The main forms of intellectual property (IP) rights are patents (inventions), trademarks (brand names), designs (the way a product looks) and copyright (for creative works).

  • Each country has a process for registration of patents, trademarks and designs. Some countries also require registration of copyright in some circumstances.

  • There is considerable uniformity across countries regarding the laws and registration processes for intellectual property rights. This is in part because most countries agree to meet obligations under international agreements such as the WTO Agreement on Trade-Related Aspects of Intellectual Property Rights (the TRIPS Agreement), which requires member nations to enact laws that meet certain minimal requirements for the protection of intellectual property rights.

  • An owner of intellectual property can sell its IP rights, or can retain its rights and license some or all of the rights for use by another party. If a licence is granted, it is usually done for a fee, called a royalty.

  • Franchising in many respects is a form of IP licensing arrangement. Three common types of franchising arrangements are business-format franchises, product franchises and production franchises.

  • When entering into these types of franchising arrangements in an overseas country the franchisor (the person owning the IP rights in the brand, product or process) can enter into franchise agreements with franchisees by way of master franchising, joint-venture franchising, area development agreements or by establishing a directly owned operation in the overseas country, by way of either a branch or a subsidiary.

  • Most countries have some form of regulation of franchising. In developed countries two of the most common forms of regulation are codes of conduct for franchisors, and competition and consumer protection laws.

  • The terms of a franchising arrangement are set out in an agreement between the franchisor and franchisee, and cover the obligations of each party.

  • 4 - Payment

  • Justin Malbon, Monash University, Victoria, Bernard Bishop, Griffith University, Queensland
  • Book:
    Australian Export
    Published online:
    06 August 2018
    Print publication:
    30 October 2014, pp 78-127

  • Summary

    Key Concepts

  • A range of risks for parties to international sale of goods transactions are in part due to the parties being in different legal jurisdictions. This can render the enforcement of contractual and other rights both complex and prohibitively expensive. There are also the risks of additional costs and delays from shipping goods across borders and, in some cases, across considerable distances. Heightened risks of non-payment and non-delivery may arise if the parties have not previously dealt with each other, and have not developed a trusted business relationship.

  • A seller of goods is at risk of not being paid by the buyer for those goods, especially if there is a delay between when the goods were shipped from the port of export and their arrival in the port of import.

  • A number of payment methods can reduce the risk for the seller of non-payment and for the buyer of non-delivery of the goods after having paid for them.

  • In some cases, banks can be used as the ‘go-between’ so as to reduce the risks of non-payment.

    • What is Covered in This Chapter

    This chapter outlines:

  • The processes involved in four main methods of payment for goods in international transactions

  • Risks associated these methods

  • How these risks can be minimised.

Page 1476 of 1880