张量方法在信赖域算法中的应用

The Application of Tensor Methods to Trust Region Algorithm

信赖域算法是求解无约束优化问题的一种有效的算法.对于该算法的子问题,本文将原来目标函数的二次模型扩展成四次张量模型,提出了一个带信赖域约束的四次张量模型优化问题的求解算法.该方法的最大特点是:不仅在张量模型的非稳定点可以得到下降方向及相应的迭代步长,而且在非局部极小值点的稳定点也可以得到下降方向及相应的迭代步长,从而在算法产生的迭代点列中存在一个子列收敛到信赖域子问题的局部极小值点.

Trust region algorithm is an effective algorithm for unconstrained optimization problems. In this paper, for the subproblem of this algorithm, we expand the objective function from twice-order model to fourth-order tensor model, then propose a new method for solving the trust-region constrained optimization problem with its objective function being a fourth-order tensor model. This method can obtain the descent direction and the corresponding steplength not only on the non-stable points but also on the stable points which are not the （local） minimum. Therefore, it is shown that, in the sequence generated by the proposed method, there must exists a subsequence which converges to the minimum of the subproblem.