A parametric class of trust-region algorithms for unconstrained nonconvex optimization is considered where the value of the objective function is never computed. The class contains a deterministic version of the first-order Adagrad method typically used for minimization of noisy function, but also allows the use of (possibly approximate) second-order information when available. The rate of convergence of methods in the class is analyzed and is shown to be identical to that known for first-order optimization methods using both function and gradients values, recovering existing results for purely-first order variants and improving the explicit dependence on problem dimension. This rate is shown to be essentially sharp. A new class of methods is also presented, for which a slightly worse and essentially sharp complexity result holds. Limited numerical experiments show that the new methods' performance may be comparable to that of standard steepest descent, despite using significantly less information, and that this performance is relatively insensitive to noise.

翻译：考虑一类用于无约束非凸优化的参数化信赖域算法，其中目标函数值从未被计算。该类算法包含通常用于噪声函数最小化的首阶Adagrad方法的确定性变体，但也允许在可用时使用（可能近似的）二阶信息。分析了该类算法的收敛速率，证明其与同时使用函数值和梯度值的首阶优化方法已知的收敛速率相同，既恢复了纯首阶变体的现有结果，又改进了对问题维度的显式依赖性。该速率被证明是本质上最优的。同时提出了一类新方法，其复杂度结果稍差且本质上最优。有限的数值实验表明，尽管新方法使用的信息显著更少，但其性能可能与标准最速下降法相当，并且该性能对噪声相对不敏感。