grams of which the objective is given by the ratio of a convex by a positive (over a convex domain) concave function. As observed by Sniedovich (Ref. [102, 103]) most of the properties of fractional pro grams could be found in other programs, given that the objective function could be written as a particular composition of functions. He called this new field C programming, standing for composite concave programming. In his seminal book on dynamic programming (Ref. ), Sniedovich shows how the study of such com positions can help tackling non-separable dynamic programs that otherwise would defeat solution. Barros and Frenk (Ref. ) developed a cutting plane algorithm capable of optimizing C-programs. More recently, this algorithm has been used by Carrizosa and Plastria to solve a global optimization problem in facility location (Ref. ). The distinction between global optimization problems (Ref. ) and generalized convex problems can sometimes be hard to establish. That is exactly the reason why so much effort has been placed into finding an exhaustive classification of the different weak forms of convexity, establishing a new definition just to satisfy some desirable property in the most general way possible. This book does not aim at all the subtleties of the different generalizations of convexity, but concentrates on the most general of them all, quasiconvex programming. Chapter 5 shows clearly where the real difficulties appear.
Table of Contents1 Introduction.- 2 Elements of Convexity.- 2.1 Generalities.- 2.2 Convex sets.- 2.2.1 Hulls.- 2.2.2 Topological properties of convex sets.- 2.2.3 Separation of convex sets.- 2.3 Convex functions.- 2.3.1 Continuity of convex functions.- 2.3.2 Lower level sets and the subdifferential.- 2.3.3 Sublinear functions and directional derivatives.- 2.3.4 Support functions and gauges.- 2.3.5 Calculus rules with subdifferentials.- 2.4 Quasiconvex functions.- 2.5 Other directional derivatives.- 3 Convex Programming.- 3.1 Introduction.- 3.2 The ellipsoid method.- 3.2.1 The one dimensional case.- 3.2.2 The multidimensional case.- 3.2.3 Improving the numerical stability.- 3.2.4 Convergence proofs.- 3.2.5 Complexity.- 3.3 Stopping criteria.- 3.3.1 Satisfaction of the stopping rules.- 3.4 Computational experience.- 4 Convexity in Location.- 4.1 Introduction.- 4.2 Measuring convex distances.- 4.3 A general model.- 4.4 A convex location model.- 4.5 Characterizing optimality.- 4.6 Checking optimality in the planar case.- 4.6.1 Solving (D).- 4.6.2 Solving (D’;).- 4.6.3 Computational results.- 4.7 Computational results.- 5 Quasiconvex Programming.- 5.1 Introduction.- 5.2 A separation oracle for quasiconvex functions.- 5.2.1 Descent directions and geometry of lower level sets.- 5.2.2 Computing an element of the normal cone.- 5.3 Easy cases.- 5.3.1 Regular functions.- 5.3.2 Another class of easy functions.- 5.4 When we meet a “bad” point.- 5.5 Convergence proof.- 5.5.1 The unconstrained quasiconvex program.- 5.5.2 The constrained quasiconvex program.- 5.6 An ellipsoid algorithm for quasiconvex programming.- 5.6.1 Ellipsoids and boxes.- 5.6.2 Constructing a localization box.- 5.6.3 New cuts.- 5.6.4 Box cuts.- 5.6.5 Parallel cuts.- 5.6.6 Modified algorithm.- 5.7 Improving the stopping criteria.- 6 Quasiconvexity in Location.- 6.1 Introduction.- 6.2 A quasiconvex location model.- 6.3 Computational results.- 7 Conclusions.