Abstract: An improved method is used to solve unconstrained optimization problems，when the derivatives of the objective function can not be available or the calculation of the derivatives are too expensive.Generally speaking，the objective function is approximated to quadratic interpolation model which is solved in a traditional trust region.The traditional trust region method is improved in this paper，and formed two novel trust region subproblems.The innovation of the improved trust region algorithm is that it solving the quadratic interpolation model in one parameterized trust region and modifying the model in another parameterized region，which can save the calculation and improve the accuracy.The improved trust region method can quickly find a descent direction and a poised interpolation point in two improved trust regions，respectively.Experimental results reveal that the improved method is more effective than the classic trust region method on the testing problems.

Key words: trust region method, quadratic interpolation model, unconstrained optimization, derivative free optimization

摘要： 一种改进的信赖域方法被用来解无约束最优化问题，当目标函数的导数信息不可利用或者求解目标函数的导数代价太大。通常，考虑用二次插值模型来逼近目标函数，并且用传统的信赖域方法求解这个二次模型。传统的信赖域方法将被改进，并且形成两个改进的信赖域子问题。改进的信赖域方法的创新点在于：求解二次模型在一个参数化的信赖域中，修改这个模型在另一个参数化的信赖域当中。在这两个新的信赖域中，可以分别很快地找到一个好的下降方向和一个具有均衡性的插值点。这个改进的方法不但节省了函数值计算次数而且提高了解的精度。实验结果表明，针对测试问题，提出的方法的确是优于传统的信赖域方法的。

关键词: 信赖域方法, 二次插值模型, 无约束最优化, 无导数最优化