The Mathematics of Optimization

Abstract

In “Introduction to Optimization Models” (UVA-QA-0682), we explored the basics of using optimization models, or mathematical programming. In this technical note, we turn our attention to the mathematics underlying optimization. While not essential, a deeper understanding of the math can help you become a significantly more effective and efficient user of optimization models. It is well worth the investment if you anticipate using optimization on a regular basis. Among other things, a deeper understanding of the math enables you to use the more advanced options that are generally available in most optimization software packages. (These options enable you, for example, to solve large models more quickly than you otherwise might.) Excerpt UVA-QA-0683 March 29, 2010 The Mathematics of Optimization In “Introduction to Optimization Models” (UVA-QA-0682), we explored the basics of using optimization models, or mathematical programming. In this technical note, we turn our attention to the mathematics underlying optimization. While not essential, adeeper understanding of the math can help you become a significantly more effective and efficient user of optimization models. It is well worth the investment if you anticipate using optimization on a regular basis. Among other things, a deeper understanding of the math enables you to use the more advanced options that are generally available in most optimization software packages. (These options enable you, for example, to solve large models more quickly than you otherwise might.) In “Introduction to Optimization Models,” we used the language of the spreadsheet to describe, build, and solve optimization models. This was useful because the spreadsheet is clearly the preferred medium for implementing quantitative business analysis of all kinds, including math programming. In this note, however, we will draw heavily on the language of algebra, particularly graphs. This was the original language of optimization models and will help us deal with some of the more technical issues. We begin by considering the algebraic statement of optimization models. Using algebra, it is easy to distinguish between the three different categories of optimization models explored in “Introduction to Optimization Models”—nonlinear, linear, and integer. We then devote a section apiece to the most commonly encountered solution techniques for each category of the program. (Remember: “Solving” an optimization model means finding an optimal solution.) Most advanced options in optimization software packages are levers that allow you to direct and control the solution procedures for the various categories of problems. Understanding the solution techniques enables you to control the levers. . . .