Practical Financial Optimization: Decision Making for Financial Engineers

Appendix A

Basics of Optimization

We give here some basic definitions on optimization, defining the optimality conditions for both linear and nonlinear programs. Readers of this book need not concern themselves with the details of solution algorithms for linear or nonlinear programs, as today robust and efficient software is widely available. The FINLIB library of Chapter 14 relieves readers of the burden of studying the technicalities of solution algorithms. However, the development of some models, and especially the analysis of some of their properties, relies on the optimality theory for optimization problems. This appendix provides adequate coverage, and additional material can be found in the textbooks by Luenberger (1984), and Nash and Sofer (1996).

A.1 Duality

There are many different ways to represent linear programs that are equivalent. Some formulations are more suitable for illustrating a property of the linear program, while others simplify the description of an algorithm. We give here what is commonly known as the “standard form” of linear programming using vector-matrix notation.

Here c and x are n-dimensional vectors of cost coefficients and decision variables, respectively; ...