Application to Global Optimization: Numerical Methods

Abstract

Numerical methods for global optimization are becoming an increasingly important field of study. There is a recognized practical need for methods which will efficiently solve global optimization problems (see, for example, Horst and Thy [81]). However, in general, such problems are, by their very nature, extremely difficult to solve. This is primarily due to the lack of tools which provide global information about the objects (sets and functions) under study in contrast with local optimization, where the classical calculus and its modern generalization provide necessary tools. It should also be noted that standard nonlinear programming methods are not generally applicable as solution methods due to the intrinsic multi-extremality of global optimization problems. Finally, the computational difficulties of global optimization through its essentially combinatorial nature make the development of general efficient solution methods unlikely. However, despite these difficulties it is possible to provide methods for the solution of some specific global optimization problems, which cover many practical needs. In this chapter we shall use techniques from abstract convexity for developing some numerical methods for global optimization.