Improved Generalized Reduced Gradient Method And Adaptive Trust Region Methods

In this paper, we will discuss and solve two problems, they are unconstrained optimization problems and constrained optimization problems.For constrained optimization problems, general reduced gradient method has been proved to be efficent for solving nonlinear programming problems, one advantage of General reduced gradient method is the dimension of the problem is reduced by variable elimination. this advantage improves the convergence speed of the algorithm.but there are also some obvious backdraws.for example, General reduced gradient method need the feasibility of every iterative point, but it is useless for Most of the time, then the speed of this algorithm is largely reduced.In this paper, basing on the General reduced gradient method,we propose a new criterion to testify whether the iterative point can be accepted or not.this criterion is basing on the line search filter technique. This new method can pursue the convergence without the feasibility of every iterative point. Thus the speed of the convergence of General reduced gradient is improved.the global convergence properties are discussed, and the numerical experiments will show this new algorithm is efficient.For unconstrained optimization problems, an adaptive trust region method with line search for unconstrained optimization problems is presented and analyzed.the trust region radius is adjusted with a new self-adaptive adjustment strategy. Numerical results show that the new method is efficient.