Non-linear (no) Constraints Conjugate Gradient Path Method

Optimization theory and methods deals with selecting the best of manypossible decisions in real-life environment, constructing computational methods tofind optimal solutions, exploring the theoretical properties, and studying thecomputational performance of numerical algorithms implemented based oncomputational methods. It is not only important in its own right but nowadays formsan integral part of a great number of applied sciences such as engineering,economics, management, industry and all branches of math-oriented engineering.The conjugate gradient method was proposed by Hestenes and Stiefel in the1950s as an iterative method for solving linear systems with positive definitecoefficient matrices. In the1960s conjugate gradient and conjugate directionmethods were extended to the optimization of nonquadratic functions. Fletcher andReeves extended the method to an optimization method for unconstrained nonlinearoptimization in1964. It is one of the earliest known techniques for solving large-scalenonlinear optimization problems. Over the years, many variants of this originalscheme have been proposed, and some are widely used in practice. The key featuresof these algorithms are that they require no matrix storage and are faster than thesteepest descent method.In this paper, we focus on use Conjugate Gradient method to get efficientand robust algorithms for solving nonlinear optimization with constrained orunconstrained. By applying the restrictively preconditioned conjugate gradient (RPCG)method to extend system of reduced preconditioned equation, we can get a morerobust and effective technique to solve large sparse system of linear equations of ablock two-by-two structure. The global convergence and local superlinearconvergence rate of the proposed algorithm are established under some reasonableconditions.The thesis consists of four parts. In Chapter1, we give a summary of thebasic concepts of optimization. In Chapter2, we propose a new approach via discreteconjugate gradient path for solving unconstrained optimization. The global convergence and local superlinear convergence rate of the proposed algorithm areestablished under some reasonable conditions. The numerical results are reported toshow the effectiveness of the proposed algorithm. In Chapter3, we propose discreterestrictively conjugate gradient path for solving linear equality constrainedoptimization and establish the relation between the restrictively preconditionedconjugate gradient method and the classical preconditioned conjugate gradientmethod to solve the reduced system of problem. Finally, we give some conclusionsand further research directions follow in Chapter4.