In this paper, we propose two trust-region algorithms for unconstrained optimization. The trust-region algorithms minimize a model of the objective function within the trust-region, next update the size of the region and then repeat the procedure to find a first-order stationary point for the objective function. The size of the trust-region at each step is very critical to the effectiveness of the algorithm, particularly for large-scale problems, because minimizing the model at each step needs the gradient and the Hessian information of the objective function. Our modified trust-region algorithms are opportunistic in the sense that they explore beyond the trust-region if the boundary of the region prevents the algorithm from accepting a more beneficial point. It occurs when there is a very good agreement between the model and the objective function on the trust-region boundary, and we can find a step outside the trust-region with smaller value of the model while at which the agreement between the model and the objective function remains good. We show that the algorithms are convergent. Initial numerical experiments show that the proposed algorithms are more efficient than the traditional trust-region algorithm for a large majority of problems in the CUTEst suite.

在本文中，我们提出了两种用于无约束优化的信任区域算法。信任区域算法将信任区域内目标函数的模型最小化，然后更新区域的大小，然后重复该过程以找到目标函数的一阶固定点。每一步的信任区大小对于算法的有效性至关重要，特别是对于大规模问题，因为在每一步最小化模型都需要目标函数的梯度和Hessian信息。从区域意义上讲，如果区域边界阻止算法接受更有利的观点，那么我们对改进的信任区域算法的探索是机会主义的。当模型与目标函数在信任区域边界上达成很好的一致性时，就会发生这种情况，并且我们可以在模型值较小的信任区域之外找到一个步骤，而此时模型与目标函数之间的一致性目标功能保持良好。我们证明了算法是收敛的。初始数值实验表明，对于CUTEst套件中的大多数问题，所提出的算法比传统的信任区域算法更有效。