There have been various algorithms designed for simulating natural evolution. This paper proposes a new simulated evolutionary computation model called the abstract evolutionary algorithm (AEA), which unifies most of the currently known evolutionary algorithms and describes the evolution as an abstract stochastic process composed of two fundamental operators: selection and evolution operators. By axiomatically characterizing the properties of the fundamental selection and evolution operators, several general convergence theorems and convergence rate estimations for the AEA are established. The established theorems are applied to a series of known evolutionary algorithms, directly yielding new convergence conditions and convergence rate estimations of various specific genetic algorithms and evolutionary strategies. The present work provides a significant step toward the establishment of a unified theory of simulated evolutionary computation.

In this paper, a new concept called nonlinear measure is introduced to quantify stability of nonlinear systems in the way similar to the matrix measure for stability of linear systems. Based on the new concept, a novel approach for stability analysis of neural networks is developed. With this approach, a series of new sufficient conditions for global and local exponential stability of Hopfield type neural networks is presented, which generalizes those existing results. By means of the introduced nonlinear measure, the exponential convergence rate of the neural networks to stable equilibrium point is estimated, and, for local stability, the attraction region of the stable equilibrium point is characterized. The developed approach can be generalized to stability analysis of other general nonlinear systems.

The Hopfield-type networks with asymmetric interconnections are studied from the standpoint of taking them as computational models. Two fundamental properties, feasibility and reliability, of the networks related to their use are established with a newly-developed convergence principle and a classification theory on energy functions. The convergence principle generalizes that previously known for symmetric networks and underlies the feasibility. The classification theory, which categorizes the traditional energy functions into regular, normal and complete ones according to their roles played in connection with the corresponding networks, implies that the reliability and high efficiency of the networks can follow respectively from the regularity and the normality of the corresponding energy functions. The theories developed have been applied to solve a classical NP-hard graph theory problem: finding the maximal independent set of a graph. Simulations demonstrate that the algorithms deduced from the asymmetric theories outperform those deduced from the symmetric theory.

A decomposition principle is developed for systematic determination of the dimensionality and the connections of Hopfield-type associative memory networks. Given a set of high dimensional prototype vectors of given memory objects, we develop decomposition algorithms to extract a set o flower dimensional key features of the pattern vectors. Every key feature can be used to build an associative memory with the lowest complexity, and more than one key feature can be simultaneously used to build networks with higher recognition accuracy. In the latter case, we further propose a "decomposed neural network" based on a new encoding scheme to reduce the network complexity. In contrast to the original Hopfield network, the decomposed networks not only increase the network's storage capacity, but also reduce the network's connection complexity from quadratic to linear growth with the network dimension. Both theoretical analysis and simulation results demonstrate that the proposed principle is powerful.