Chapter 16 - Global Optimization Concepts and Methods

Abstract

Basic concepts of deterministic and stochastic global optimization methods are described. Deterministic methods guarantee finding a global optimum for the cost function under certain assumptions. These include the covering methods, the zooming method, methods of generalized descent, and the tunneling method. Stochastic methods use random numbers and probabilistic ideas in their calculations. These include pure random search, multistart methods, clustering methods, controlled random search (the Nelder–Mead method), acceptance–rejection methods and stochastic integration. Some of the methods try to find all the local minima while others skip some of the local minima in search for a global minimum. Since there are no optimality conditions for a global minimum point, the methods do not guarantee global optimality of the final solution. Also, a precise stopping criterion for the algorithm cannot be given; therefore, the global minimum point cannot be recognized even if it has been found during numerical search. Two new stochastic global optimization algorithms are presented: domain elimination and stochastic zooming. Numerical performances of these methods are discussed. It is noted that most of the methods take a considerable amount of computational time to find the final solution.