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© 2018

Practical Mathematical Optimization

Basic Optimization Theory and Gradient-Based Algorithms

  • Guides readers to understand processes and strategies in real world optimization problems

  • Contains new material on gradient-based methods, algorithm implementation via Python, and basic optimization principles

  • Covers fundamental optimization concepts and definitions, search techniques for unconstrained minimization and standard methods for constrained optimization

  • Includes example problems and exercises

Textbook

Part of the Springer Optimization and Its Applications book series (SOIA, volume 133)

Table of contents

  1. Front Matter
    Pages i-xxvi
  2. Basic optimization theory

    1. Front Matter
      Pages 1-1
    2. Jan A. Snyman, Daniel N. Wilke
      Pages 3-40
    3. Jan A. Snyman, Daniel N. Wilke
      Pages 41-69
    4. Jan A. Snyman, Daniel N. Wilke
      Pages 71-112
    5. Jan A. Snyman, Daniel N. Wilke
      Pages 113-167
    6. Jan A. Snyman, Daniel N. Wilke
      Pages 169-193
  3. Gradient-based algorithms

    1. Front Matter
      Pages 195-195
    2. Jan A. Snyman, Daniel N. Wilke
      Pages 197-250
    3. Jan A. Snyman, Daniel N. Wilke
      Pages 251-271
    4. Jan A. Snyman, Daniel N. Wilke
      Pages 273-310
    5. Jan A. Snyman, Daniel N. Wilke
      Pages 311-340
  4. Back Matter
    Pages 341-372

About this book

Introduction

This textbook presents a wide range of tools for a course in mathematical optimization for upper undergraduate and graduate students in mathematics, engineering, computer science, and other applied sciences.  Basic optimization principles are presented with emphasis on gradient-based numerical optimization strategies and algorithms for solving both smooth and noisy discontinuous optimization problems. Attention is also paid to the difficulties of expense of function evaluations and the existence of multiple minima that often unnecessarily inhibit the use of gradient-based methods. This second edition addresses further advancements of gradient-only optimization strategies to handle discontinuities in objective functions. New chapters discuss the construction of surrogate models as well as new gradient-only solution strategies and numerical optimization using Python. A special Python module is electronically available (via springerlink) that makes the new algorithms featured in the text easily accessible and directly applicable. Numerical examples and exercises are included to encourage senior- to graduate-level students to plan, execute, and reflect on numerical investigations. By gaining a deep understanding of the conceptual material presented, students, scientists, and engineers will be  able to develop systematic and scientific numerical investigative skills.

 

Keywords

Mathematica algorithms linear optimization optimization programming Python multi-modal optimization non-smooth optimization discontinuous optimization Numerical Linear Algebra Hessian matrix approximations Gradient-only solution strategies Karush-Kuhn-Tucker theory Quadratic programming line search descent algorithm for unconstrained minimization Unconstrained one-dimensional minimization

Authors and affiliations

  1. 1.Department of Mechanical and Aeronautical EngineeringUniversity of PretoriaPretoriaSouth Africa
  2. 2.Department of Mechanical and Aeronautical EngineeringUniversity of PretoriaPretoriaSouth Africa

About the authors

Jan A. Snyman currently holds the position of emeritus professor in the Department of Mechanical and Aeronautical Engineering of the University of Pretoria, having retired as full professor in 2005. He has taught physics, mathematics and engineering mechanics to science and engineering students at undergraduate and postgraduate level, and has supervised the theses of 26 Masters and 8 PhD students. His research mainly concerns the development of gradient-based trajectory optimization algorithms for solving noisy and multi-modal problems, and their application in approximation methodologies for the optimal design of engineering systems. He has authored or co-authored 89 research articles in peer-reviewed journals as well as numerous papers in international conference proceedings.

Daniel N. Wilke is a senior lecturer in the Department of Mechanical and Aeronautical Engineering of the University of Pretoria.   He teaches computer programming, mathematical programming and computational mechanics to science and engineering students at undergraduate and postgraduate level. His current research focuses on the development of interactive design optimization technologies, and enabling statistical learning (artificial intelligence) application layers, for industrial processes and engineering design. He has co-authored over 50 peer-reviewed journal articles and full length conference papers.

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