Nonlinear Constrained Optimization Problem Of Affine Projection Is Reduced Hessian Modified Gradient Path Interior Point Method

Optimization has the wide applications in many fields of national economy such as nationaldefence, industrial and agricultural production, tra?c and transportation, finance, trade, manage-ment and scientific research.Nonlinear programming is the most common optimization. The problem is named constrainedoptimization if the variables are subject to some conditions, vice versa, the problem is namedunconstrained optimization if the variables have no any constraint condition. This thesis is focusedon a nonlinear programming subject to an equality constraint and boundary constraint.The optimal conditions are the conditions that the optimum solution (local or global) mustsatisfy, among which first-order necessary condition and second-order necessary condition are mostcommon. The optimal condition is not only of great significance for the research of optimizationtheory, but also plays an important role in determining the design of the algorithm and the decisionof termination condition, so we first construct a quadratic approximation in terms of the optimalconditions, and then propose a nonmonotonic reduced projected Hessian method via an a?nescaling interior modified gradient path method.Coleman and Li propose a trust region interior algorithm named "double-trust regionmethod"in [5]. This algorithm construct an a?ne scaling matrix which is an innovated andinstructive highlight. In the double-trust region method, the iterated direction will definitely lie inthe strict feasible region after appropriate trust region radius is taken by the aid of the a?ne interiortransform, which allows us to ignore the boundary constraint in solving trust region subproblem. Inthe thesis, we construct a corresponding a?ne scaling matrix according to the optimal conditionsso as to overcome the di?cult of boundary constraint. And we take reduced Hessian algorithm tosolve another di?cult of "equality constraint". Reduced Hessian algorithm has become one of themost popular and e?ective methods for solving nonlinear equality constrained programming. Prof.Zhu succeeded in solving linear equation constraint optimization by reduced Hessian trust regiontechnique in [20]. The idea of reduced Hessian, which uses the QR decomposition of the matrix inaffine equation constraint, divides the origin problem into two general subproblems in the rangesubspace and null subspace, which is of great value to the numerical solution and realization.Both line search and trust region are well-accepted methods in the nonlinear programmingto assure global convergence. The trust region method is a common and feasible method forboth unconstrained optimization and constrained optimization. The basic steps of trust region areas follows: First, we set trust-region radius and find the quadratic approximation of the objectfunction. And then we get a trial step produced by minimizing the quadratic approximation inthe trust region. The trust region method assures the global convergence of the algorithm, and itdoes not require the Hessian (or its approximation) to be indefinite. The trust region method can...