Nondifferentiable Optimization

Abstract

Scientific or engineering applications usually require solving mathematical problems. Such applications in accordance with networks span a wide range, from modeling the evolution of species in biology to modeling soap films for grids of wires; from the design of collections of data to the design of heating or airconditioning systems in buildings; and from the creation of oil and gas pipelines to the creation of communication networks, road and railway lines. These are all network design problems of significant importance and nontrivial complexity. The network topology and design characteristics of these systems are classical examples of optimization problems. I. Intuitively speaking, a network is a set of points and a set of connections where each connection joins one point to another and has a certain length. The combinatorial structure of such a network is described as a graph G which is defined to be a pair (V, E) where • V is any finite set of elements, called vertices, and • E is a finite family of elements which are un-ordered pairs of vertices, called edges. Additionally, assume that a function l: E-+ R is given for the edges of the graph G. Usually, assume that l has only positive values and call it a length-function. A (connected) graph equipped with a length-function is called a network.