| Optimization, as a powerful solution approach, finds wide applications in extremely broad spectra of fields and human activities including engineering, science, finance, etc. The existence of multiple local minima of a general nonconvex objective function, f, makes global optimization (GO) a great challenge. The main purpose of this research is to develop some high performance methods for solving global optimization problems (GOP) over continuous or discrete variables. This research makes several contributions to advance the state-of-the-art of GO. These contributions include both theoretical and numerical aspects as stated below.; In solving continuous GOP, we move from one minimizer to another better one with the help of some auxiliary functions, namely the filled functions (FF). We have proposed four novel FF's in Part I of this dissertation. The proposed FF's not only preserve the promising theoretical properties of the traditional FF but also overcome the difficulties in its computational implementation. Furthermore, the final one is superior to all the existing FF's, since it has a minimizer over the problem domain instead of a line. This property ensures that a better minimizer of f can be found by some classical local search methods. Extensive numerical experiments on several test problems with up to 1000 variables are reported. These results indicate that the proposed solution algorithms are efficient.; In Part II of this dissertation, we have developed two discrete filled function (DFF) methods to tackle discrete GOP. These methods are probably the first in discrete optimization that are capable of solving nonlinear, inseparable and nonconvex large scale integer optimization problems. The first DFF has a discrete minimizer on a “discrete path” between the current discrete minimizer and a better discrete minimizer of f. The second one not only guarantees to have a discrete minimizer over the problem domain, but also ensures that its discrete minimizers coincide with the better discrete minimizers of f. This property assures that a better discrete minimizer of f can be found by some classical local search methods. Numerical experiments on several test problems with up to 100 variables have demonstrated the applicability and efficiency of the proposed methods. |
