标题： 基于类似泰勒模型的非光滑优化：误差界、收敛性和终止准则 显示英文标题

标题： Nonsmooth optimization using Taylor-like models: error bounds, convergence, and termination criteria

摘要： 我们考虑依次最小化目标函数的简单泰勒类似模型的优化算法。用于最小化凸函数和光滑映射的复合函数的高斯-牛顿型方法是常见的例子。我们的主要结果是任何此类算法的步长与附近点处函数斜率之间的显式关系。因此，我们（1）证明步长可以可靠地用于终止算法，（2）证明只要步长趋于零，迭代点的每个极限点都是驻点，并（3）证明类似于经典二次增长的条件意味着步长线性地界定了迭代点到解集的距离。后者所谓的误差界性质通常用于建立线性（或更快）收敛保证。当步长被模型值减少量的平方根代替时，类似的结论也成立。我们通过当模型仅被不精确地最小化时的扩展来完成本文。 显示英文摘要

摘要： We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result is an explicit relationship between the step-size of any such algorithm and the slope of the function at a nearby point. Consequently, we (1) show that the step-sizes can be reliably used to terminate the algorithm, (2) prove that as long as the step-sizes tend to zero, every limit point of the iterates is stationary, and (3) show that conditions, akin to classical quadratic growth, imply that the step-sizes linearly bound the distance of the iterates to the solution set. The latter so-called error bound property is typically used to establish linear (or faster) convergence guarantees. Analogous results hold when the step-size is replaced by the square root of the decrease in the model's value. We complete the paper with extensions to when the models are minimized only inexactly.