Research On Conic Model Methods For Solving Nonlinear Equations

With the development of science and the extensive applications of the com-puter,the system of nonlinear equations has been paid more and more atten-tion.The problem of solving nonlinear equations has become an heated research topic,because it has wide applications in artificial intelligence,machine learn-ing,financial computing,disaster prevention research,energy detection,weather forecasting,etc.In this dissertation,we primarily study conic model methods for solving non-linear equations which include three aspects.The first is to solve smooth nonlinear equations by an improved conic model Newton method.The second is a structured quasi-Newton method for an unconstrained optimization with special structure of the objective function.The third is to solve a class of non-smooth equations by a smooth method.The main results obtained in this dissertation are summarized as follows:1.The relationship between the two-point rational approximation model algo-rithm and the cone model algorithm is analyzed.It is shown that the two-point rational approximation model algorithm is a special case of the cone model algorithm which provides a theoretical framework for the improve-ment and perfection of the two-point rational approximation algorithm.2.Some improved methods of the two-point rational approximation model are proposed.First,a method of selecting a more proper solution of rational approximations is presented.Next,the rational approximation is proven to be a monotonic function.Then,for having opposite signs of two direc-tional derivatives along the direction connecting the current iteration point and the previous iteration point of an original function,we propose an it-erative method of solving its rational approximation and an approximate method to construct a rational function with same sign directional deriva-tives along the direction connecting the current iteration point and another estimate point.Finally,a non-monotonic rational approximation function is presented.Numerical experimental results show the validity and feasibility of these improved methods.3.A special kind of conic model for approximating vector valued functions is proposed.We present an iterative scheme for solving nonlinear equations.Based on this model,an improved conic model Newton algorithm for solving nonlinear equations is obtained.The main feature of this algorithm is that the new method revises the Jacobian matrix by a rank one matrix in each iteration.Under general conditions we determine that the algorithm has the property of local quadratic convergence.The numerical performance and comparison show that the proposed method is efficient.4.A structured conic quasi-Newton method for special structured unconstrained optimization problems is proposed.First,the conic quasi-Newton equation is deduced by the conic model and interpolation conditions.The standard quasi-Newton equation employs only the gradients,but ignores the informa-tion of function values.The conic quasi-Newton method not only uses the in-formation of the gradients but also of the function values.Next,a structured conic quasi-Newton algorithm is proposed based on the conic quasi-Newton equation.The algorithm is suitable for solving the unconstrained optimiza-tion problem in which the Hessian of the objective function has some special structure and is partially available.The nonlinear least-squares problem is a typical example of this kind of problem.Furthermore,local and superlinear convergence of the algorithm is obtained under some reasonable conditions.5.A smoothing type algorithm for absolute value equations is proposed,and numerical comparisons are done based on four smoothing functions.The sys-tem of absolute value equations,denoted by AVE,is a non-differentiable NP-hard problem.In this paper,we rewrite the AVE as a system of smooth equa-tions and propose four new smoothing functions along with a smoothing-type algorithm to solve the system of equations.The main contribution of our work focuses on numerical comparisons which suggest a better choice of the smoothing function along with the smoothing-type algorithm.The numeri-cal performance in terms of number of iterations and computing time is also provided.