Multi-Objective Optimization Methods Based on Artificial Neural Networks

During the last years, several optimization algorithms have been presented and widely investigated in literature, most of which based on deterministic or stochastic methods, in order to solve optimization problems with multiple objectives that conflict with each other. Some multi-objective stochastic optimizers have been developed, based on local or global search methods, in order to solve optimal design problems. Despite the significant progress obtained in this field, there are still many open issues. In fact, both the deterministic and stochastic approaches present hard limits. In the first case, although the number of function evaluations needed to reach the optimal solution is generally small, the risk to be trapped in local minima is very high, whereas in the second case, the probability to reach the optimal solution is higher but the computational cost could become prohibitive. In particular, this is the case of the electromagnetic problems. Electromagnetic devices are fundamental in the modern society. They are used for storing and converting energy (Magele, 1996), manufacturing processes (Takahashi et al., 1996), magnetic resonance imaging (Gottvald et al., 1992), telecommunications, etc. The design optimization of the electromagnetic devices is one key to enhance product quality and manufacturing efficiency. Definition of geometric boundaries to achieve specific design goals together with nonlinear behaviour of ferromagnetic materials often give rise to multimodal, non-linear, and non-derivable objective functions. For this reason, resorting to numerical approaches, such as the Finite Element Method (FEM), to evaluate objective functions in many cases is compulsory. When the number of design parameters to be optimized is considerable, the number of objectives evaluations to be performed could be of the order of thousand and the use of numerical solution during the optimization process can be unfeasible. Approximating techniques have been proposed as a way to overcome the time consuming numerical procedure (Alotto et al., 2001, Canova et al., 2003, Wang & Lowther, 2006). One of the most effective approximation approaches is based on Artificial Neural Networks. In fact, an alternative method to numerical evaluation consists of applying the optimization procedure to the approximation of the objective function, rather than to its numerical model (Abbass, 2003, Fieldsend & Singh, 2005, Carcangiu et al., 2008). On the other hand, the quality of the solution of the optimization problem depends on the error introduced by the approximation