Quadratic Programming with Computer Programs
eBook - ePub

Quadratic Programming with Computer Programs

  1. 386 pages
  2. English
  3. ePUB (mobile friendly)
  4. Available on iOS & Android
eBook - ePub

Quadratic Programming with Computer Programs

About this book

Quadratic programming is a mathematical technique that allows for the optimization of a quadratic function in several variables. QP is a subset of Operations Research and is the next higher lever of sophistication than Linear Programming. It is a key mathematical tool in Portfolio Optimization and structural plasticity. This is useful in Civil Engineering as well as Statistics.

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Yes, you can access Quadratic Programming with Computer Programs by Michael J. Best in PDF and/or ePUB format, as well as other popular books in Business & Operations. We have over one million books available in our catalogue for you to explore.

Information

Publisher
Chapman and Hall/CRC
Year
2017
eBook ISBN
9781351647205
Topic
Business
Subtopic
Operations
Index
Business

Chapter 1

Geometrical Examples

The purpose of this chapter is to introduce the most important properties of quadratic programming by means of geometrical examples. In Section 1.1 we show how to determine an optimal solution geometrically. A major difference between linear and quadratic programming problems is the number of constraints active at an optimal solution and this is illustrated by means of several geometrical examples. Optimality conditions are derived from geometrical considerations in Section 1.2. The geometry of quadratic functions is developed in terms of eigenvectors and eigenvalues in Section 1.3. Nonconvex quadratic programming problems are introduced geometrically in Section 1.4 where it is shown that such problems may possess many local minima.

1.1 Geometry of a QP: Examples

We begin our analysis of a quadratic programming problem by observing properties of an optimal solution in several examples.
Example 1.1
minimize:−4x1−10x2+x12+x22subject to:−x1+x2≤2,(1)x1+x2≤6,(2)x1≤5,(3)−x2≤0,(4)−x1≤−1.(5)
The objective function1 for this problem is f(x)=−4x1−10x2+x12+x22.
The feasible region, denoted by R, is the set of points
x=[x1x2]
which simultaneously satisfy constraints (1) to (5). An optimal solution is a feasible point for which the objective function is smallest among those points in R. The feasible region has the same form as that for a linear programming problem. The difference between a quadratic and a linear programming problem is that the former has a quadratic objective function while the latter has a linear objective function. A linear programming problem is thus a special case of a quadratic programming problem in which the coefficie...

Table of contents

  1. Cover Page
  2. Half title
  3. Title Page
  4. Copyright Page
  5. Advances in Applied Mathematics
  6. Dedication Page
  7. Table of Contents
  8. Preface
  9. Using the Matlab Programs
  10. 1 Geometrical Examples
  11. 2 Portfolio Optimization
  12. 3 QP Subject to Linear Equality Constraints
  13. 4 Quadratic Programming Theory
  14. 5 QP Solution Algorithms
  15. 6 A Dual QP Algorithm
  16. 7 General QP and Parametric QP Algorithms
  17. 8 Simplex Method for QP and PQP
  18. 9 Nonconvex Quadratic Programming
  19. Bibliography
  20. Index