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Convergence Analysis Of Some Algorithms For Equilibrium Problem And Optimization Problem

Posted on:2014-02-05Degree:DoctorType:Dissertation
Country:ChinaCandidate:J F BaoFull Text:PDF
GTID:1260330428959267Subject:Computational Mathematics
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In this work, we propose some exact and inexact algorithms for solving Equilibrium problem (EP) and nonlinear least squares problem. Under certain conditions, we investigate the convergence properties of these algorithms. The main work is organized as follows.In Chapter2, we propose four algorithms for EP by combining projection method and proximal-like method. These two-steps algorithms are all devised for solving EP by the Convex feasibility problem. In view of computation, these algorithms are all proposed by combining the proximal-like method and the most violated constraint control strategy to simplify the nonlinear level set constraint of projection to the original set of EP. Mean-while, the proposed algorithms require weaker conditions compared with the proximal-like algorithm, which provides another choice for solving EP. Under some certain conditions, we investigate the convergence properties of these algorithms. Additionally, under the con-dition that the error bound holds for the related inequality system, we establish the linear convergence of the algorithm. At last, we take a variational inequality problem and a gen-eral equilibrium problem as examples for the numerical experiment, with the results show the effectiveness of the algorithms.We also demonstrate by comparison the suitable region of the parameters and the robustness of the algorithms.In Chapter3, we propose three approximate Gauss-Newton algorithms for the non-linear least squares problem under the assumption that the Jacobian matrix is not of full column rank. Since the solution of the normal equation is not unique, we assume the Jaco-bian matrix is of full row rank to guarantee by the property of the pseudo-inverse that the solution with least norm can be found while doing the inexact computation for the large-scale problem. With proper residuals control, we set up the Kantorovich-type theorem by constructing a majorizing function, and obtain a local convergence theorem including the convergence radius. Furthermore, under the assumption that the residuals satisfying the second order control, we get the quadratic convergence Kantorovich-type theorem and an explicit criterion. Numerical examples and the comparison with other papers show the ef-fectiveness of the algorithms and the criteria.
Keywords/Search Tags:Equilibrium problem, Convex feasibility problem, Error bound, Projec-tion method, Proximal point method, Nonlinear least squares problem, Approximate Gauss-Newton method, Kantorovich-type criterion
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