Solving Nonlinear Systems And Global Optimization With Interval Algorithm

Interval analysis is a new branch of numerical analysis. It has been applied tovarious field. This aricle presents an in-depth and detailed study of interval iterationalgorithm of nonlinear equation and interval global optimization algorithms. In additionto a carefull analysis of the strengths and weaknesses of interval algorithms, someimproved iteration and optimization algorithms are presented and the satisfactory resultsare achieved. The main work of the dissertation can be summarized as follows:The fundations of interval analysis including the background, basic notations, andinterval functional expansion are introduced. Meanwhile, the overview of reserch athome and abroad nowadays are given. Later, we introduce the basic interval iterationalgorithm in soving nonlinear equation sets and the interval optimization methodologyin dealing with global optimization and analyze the strenghs and weaknesses of thesecurrent algorithms.Considering the dependency problems , we present an improved algorithm basedon interval and affine arithmetic to solve nonlinear equations. This algorithm can keeptrack of first-order correlations between computed and inputs quantities, and thesecorrelations are automatically exploited in primitive operation. This method can solvesome nonlinear equations effectively. Results indicate that the iteration times can besubstantially reduced and a safe staring region can be wider.After introducing the concept of genetic algorithm, we propose the hybridalgorithms based genetic algorithm and interval method, which can utilize theadwantages of both algorithms.When solving nonlinear equations, the problem is transformed into that of functionoptimization. A new interval-genetic algorithm (IGA) is presented, which combinesgenetic algorithm and interval algorithm. The algorithm has the advantages of thegenetic algorithm such as group search and global convergence and the intervalalgorithm such as the special computational test for the existence of a solution. At eachiteration the interval algorithm provides the reliable domain for the genetic algorithm tosearch, and the genetic algorithm gives the safe starting regions for the intervalalgorithm. Finally, numerical experiments show that the IGA has globally convergence,high convergence rate and solution precision, and is a reliable approach in solvingnonlinear equations.When solving global optimization of functions, a hybrid interval genetic algorithm(HIGA)is presented too. This algorithm uses the interval branch-and-bound principle toobtain small and reliable regions where candidate solutions lie. In this way, ahighly-performing initial population is formed for genetic algorithm. Furthermore, thegenetic algorithm using this population gives upper bound of global optimum used to discard the intervals. Finally, numerical experiments show that HIGA outperformstraditional interval algorithm and genetic algorithm.