Overview

Linear programming models cover numerous applications. Whenever applicable, LP allows one to obtain useful quantitative and qualitative information on the problem at hand. The specific analytic structure of LP programs gives rise to a number of general results (e.g., those of the LP duality theory) that provide us in many cases with valuable insight and understanding (see, e.g., Exercise 1.19). At the same time, this analytic structure underlies some specific computational techniques for LP; these techniques, which by now are perfectly well developed, allow one to solve routinely quite large (tens or hundreds of thousands of variables and constraints) LP programs. Nevertheless, there are real-life situations that cannot be covered by LP models. To handle these essentially nonlinear cases, one needs to extend the basic theoretical results and computational techniques known for LP beyond the bounds of LP.

For the time being, the widest class of optimization problems to which the basic results of LP were extended is the class of convex optimization programs. There are several equivalent ways to define a general convex optimization problem; the one we are about to use is not the traditional one, but it is well suited to encompass the range of applications covered in this book.

When passing from a generic LP problem

to its nonlinear extensions, we should expect to encounter some nonlinear components in the problem. The traditional way here is to say, "Well, in (LP) there are a linear objective function fo( x) = c ^T