Affine Linear Inequality Constraints Semismooth System Interior Point Trust Region Method

Optimization theory and method which is a very extensive applieddisciplines, along with the popularization of computer informationtechnology,numerical optimization method has been developingquickly which is used in many fields, such as defense,industrialproduction, transportation,finance, economic plan,engineering design,production management, etc. At present, the problem of solving thenonlinear semismooth equations effectively has become importantaspects in investigation, and to solve the linear inequality constraintsnonlinear semismooth equations that is mainly transformed intoequivalent constraint optimization problem, then solve it by combinedwith mature optimization method. Currently, there were many articleswhich researched the solutions of the linear inequality constrainedsemismooth systems. However, the methods for solving the linearinequality constrained semismooth systems were very limited. Actually,the applications of the linear inequality constraints case are wider inpractice. So the paper proposed affine interior point trust regionalgorithm for solving the linear inequality constrained semismoothsystems.This paper will transform the linear inequality constraints semismoothsystem into equivalent optimal problem, and build trust regionsubproblems. We obtain Neton step by solving semismoothGauss-Neton equation, then get projection Newton step in the feasibleregion. In order to ensure that the loss of the objective function fully inthe algorithm iteration in the process, we can get best iteration stepwith the related properties of cauchy step. In this paper, considering theproblem of inequality constraints, we combine the affine interior pointtrust region method with the non-monotone gradient projectionstrategy to solve the linear inequality constrained semismooth systems. The global convergence and superlinear convergence rate of theproposed algorithm areestablished under suitable assumptions. The numerical experimentshave explained feasibility and effectiveness of the algorithm. The paperconsists of two parts. In Chapter1, we briefly review the basic conceptof the optimization. In Chapter2, we present the algorithm of affineinterior point trust region for solving the linear inequality constrainedsemismooth. Then the properties of Global convergence and localsuper linear convergence rate are proved under some reasonableconditions. Furthermore, we give some relative numerical results whichindicate the effectiveness of the algorithm by applying themathematical software Matlab.