Quasiconvex Optimization and Location Theory

1 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.