Research On Trust Region Methods Based On The Fractional Model

Trust region method is a kind of important numerical calculation method for nonlinear optimization.The method has good stability and strong convergence.Traditional trust region methods are based on a quadratic model.However,some objective functions in practical problem may have non-quadratic behavior or its curvature changes severely,the quadratic model methods often produce a poor prediction of the minimizer of the function.Therefore,non-quadratic models which incorporate more interpolation information in the iterations and improve the behavior of optimization algorithms are built up.Conic model proposed by Davidon can take into account more information from previous iterations than quadratic model.Using the conic model to approximate the original function may be better than the quadratic model,but there is only one horizontal vector parameter,which affects the choice of the search direction.Therefore,we consider the extended form of the quadratic model and conic model-fractional model.It contains three horizontal vectors.By making full use of the function information of the previous iteration process,we can select the three horizontal vectors to satisfy more interpolation conditions,so as to better approximate the original objective function.When the iterative point approaches the minimum point,the fractional model degenerates into a quadratic model.Thus the fractional model preserves the advantage that the quadratic model converges quickly near the minimum point.This paper mainly studies the fractional model trust region algorithm and its application in optimization problems.First,we propose a new model with three parameter vectors—the fractional model,simplify the trust region subproblem of the fractional model by selecting the control parameters,and solve it in the quasi-Newton direction.Then,a fractional model trust region quasi-Newton algorithm for solving unconstrained optimization problems is proposed.Based on this,we study in depth on the Newton point and the steepest descent point of the fractional trust region subproblem,so as to construct a simple polyline method to solve the subproblem.Numerical experiments show that the fractional model trust region quasi-Newton algorithm seems to be superior to the conic model trust region algorithm in terms of the number of iterations and the running time as the dimension of the optimization problem increases.For the linear equality constrained optimization problem,the null space technique is used to delete the linear equality constraint,and the fractional trust region method for solving the linear equality constrained optimization problem is proposed.Second,we also use the fractional model to solve the nonlinear equality constrained optimization problem.By cyclically fixing the fractional coefficient part of the fractional approximation function,the nonlinear equality constraint trust region subproblem with the fractional model is transformed into a simple one-dimensional quadratic model subproblem,and a new approximate solution method for solving the subproblem is obtained.Then,we propose a quasi-Newton algorithm for the nonlinear equality constraint problem based on the new solution to the subproblem.The convergence of the algorithm is proved.The numerical results show that the new algorithm is more stable and effective.Last,for the unconstrained optimization problem,inspired by the alternating direction multiplier method,we search for an approximate solution to the new conic trust region subproblem in two mutually orthogonal directions,and then a new conic model trust region algorithm based on alternating direction search method for the unconstrained optimization problem is proposed.The numerical results show that the new algorithm is superior to the algorithm for solving the new conic trust region subproblem by the dogleg method.On this basis,we also consider the nature of the descending direction of the parameter vector,so the calculation is simplified by adding the assumptions.The new modified algorithm is simple to calculate,and has better numerical results.