Dos Santos Gromicho J.A. Quasiconvex Optimization and Location Theory

Springer, 1998. — 233 p.A crucial step in making decisions of a quantitative nature requires the solution of optimization problems. Such optimization problems can be described as minimizing or maximizing an objective function subject to a family of constraints. The choice of which functions to use as an objective function and as constraints depends on the modeling of the problem. However, modeling a problem is only useful in practice if there exist methods to solve the proposed model. This justifies the enormous popularity of linear programs, i.e. optimization problems where the objective function and the constraints are linear functions. In fact, a practical method, the popular simplex method, exists since the late forties to solve these problems. In spite of this, there are many real life situations which cannot be appropriately modeled using only linear functions and therefore the field of nonlinear programming has received increasing attention from many researchers.
Nowadays, convex analysis, which started in a modern sense with the well-known book of Rockafellar, is recognized as the main analytical tool for analyzing nonlinear optimization programs. This led to the creation of a new field called convex programming. Clearly, convex programs are more flexible from a modeling point of view than linear programs and form an important subclass within nonlinear programming. The books of Hiriart-Urruty and Lemankhal serve as excellent references for results on convex programs. A short overview without proofs of some results in convex analysis is given in Chapter 2 of this book.
One of the most important characteristics of convex programs is that differentiability of the objective or constraint functions is not required. In fact, gradients are replaced by subgradients. Shor emphasizes this relation by using the term non-differentiable optimization as a synonym for convex programming.Front Matter
Introduction
Elements of Convexity
Convex Programming
Convexity in Location
Quasiconvex Programming
Quasiconvexity in Location
Conclusions
Back Matter