Intelligent Optimization Algorithms For Solving Nonlinear Equations

With the rapid development of science and technology, nonlinear equations play an importantrole in many fields of mechanics, circuits, power systems, nonlinear differential equations andso on. Solving nonlinear equations is still a problem at present because of the lack of reliableand efficient algorithm, especially to nonsmooth equations and high nonlinearity practicalengineering problems. In spite of that there are lots of traditional solutions of nonlinearequations, these methods have a higher intrinsic requirement for the equation. Which not onlyto meet conductivity requirements, but also depend on the selection of the initial point, andtherefore they have many limitations. We can avoid the problems such as the objectivefunction not differentiable and hard to select initial point in the non-linear equation solving bybiological heuristic intelligent optimization algorithms. This paper introduces intelligentoptimization algorithms for solving nonlinear equations based on Particle SwarmOptimization algorithm and Brain Storm Optimization algorithm.Particle Swarm Optimization algorithm and Brain Storm Optimization algorithm arereviewed Firstly. The core of Particle Swarm Optimization algorithm is the updating formulaabout the speed and position of the particle. The particles update their velocity and position bythe updating formula. And then this paper reports on the improvement in some parameters.With the addition of Inertia weight, compression factor and other parameters, the performanceof the algorithm is improved in the search ability and the convergence.Brain Storm Optimization algorithm simulates group behavior of human being in theprocess of solving problems. It produces as many new individuals as possible by threeoperations of grouping, replacing, and creating, so as to find the best individual in theincessant generation. In the improvement of Brain Storm Optimization algorithm, a simplergrouping method called SGM takes the place of the k-mean clustering method in thegrouping operation, and the policy named IDS is used to produce the random noise value inthe grouping operation. These improvements reduce the computational complexity andimprove the search ability of the algorithm.Consider the systems of nonlinear equations where f_j(j=1,2,...m)is a real-valued function on area D of the n-dimensional EuclideanspaceR n,and they are at least one of which is a nonlinear function. In order to applyintelligent optimization algorithms for solving nonlinear equations, the problem of solvingnonlinear equations is transformed into a nonlinear least squares problem, formulated as afunction optimization problem:F(x) is used as an evaluation function of adaptive degree. In general, the iteration stopswhen the fitness value meets the accuracy requirement within the maximum number ofiterations, then output the solution about systems of nonlinear equations.These intelligent optimization algorithms overcome the problem that initial point is hardto be selected and the target function is not differentiable in the traditional method. The testresults show that these algorithms are effective and feasible from results of five systems ofnonlinear equations.