Because of the rapid development of computer technology some tasks that were impossible to be solved in the past can be accomplished with computer. The overlapping and development between life science and engineering is a vivid characteristic in modern science and technology. It is also a research focus of relative fields. There are still many issues in modern scientific theory and practice with regard to Combination & Optimization and self-adaptation etc. The routine methods are helpful to resolve simple optimization and self-adaptation problems but helpless for complicated large-scale systems. So people devised Genetic Algorithm by simulating the genetic and evolution mechanism of biology. As Genetic Algorithm has vivid biological character and is suitable for any functions, it has broad application. Studying the algorithm has great significance.Genetic Algorithms, based on the biological mechanism of natural selection & heredity and leveraging colony searching technology, is particularly applicable for the solution of complicate & non-linear problems which are intractable with traditional searching methods. For nearly 40 years' development Genetic Algorithms has made great achievements in both theory research and practical applications. However, its mathematical foundation is still incomplete compared with the distinctive and sound biological foundation.This paper starts with the basic theory of Genetic Algorithms. Then some modification methods are advanced, validity analysis is carried on and some convergence proofs are made, aiming at the early-maturing phenomena of Simple Genetic Algorithm (SGA) and the problem that the probability converged to optimal solution is less than 1. The major tasks in this paper include:(1) Improved traditional crossover operators, proposed the uniform block crossover operator which could overcome the early-maturing phenomena effectively; (2) For non-convergence by probability 1 and the control drawback of the traditional Genetic Algorithm, proposed Improved Genetic Algorithm(IGA) with the synthetically control strategy;(3) The valid analysis of the uniform block crossover operators was carried on in theory. We demonstrated that their solutions are valid through real function optimization and TSP problem;(4) Made convergence analysis of IGA using the principle of Contractive Mapping in the function analysis theory; (5) Using the Genetic Algorithm solved a difficult problem - the best uniform approximation in numerical analysis, and widened it into the approach questions with other norms.Through the improvement to the Simple Genetic Algorithm, the efficiency of Genetic Algorithm has advanced, and it is proved in theory and in practice. We have got a general program solving the normal approximate problems through solving the best uniform approximation. We think that this program has important significance in practice. |