分段光滑优化中的强增长与次梯度
Strong growth and Goldstein subgradients in piecewise smooth optimization
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中文总结 AI 辅助
该研究针对分段光滑优化问题，探讨了Lipschitz函数局部极小值点附近的新增长条件，明确了强凸与非凸情形下该条件的成立情况，并通过计算实验验证了假设的重要性。
中文摘要 AI 辅助
我们研究了Lipschitz函数在局部极小值点附近的一种新增长条件，该条件是近期在基于Goldstein次梯度的算法及其实际性能的背景下提出的。在通用示例中，常观察到这些算法近似线性收敛。我们聚焦于分段二次连续可微的目标函数，对于这类函数，当目标函数额外为强凸时，该新增长条件成立；在非凸情形下，我们证明二次增长结合有效梯度的正则性属性即可满足该条件。计算实验验证了这些假设的重要性。
英文摘要
We explore a new growth condition for Lipschitz functions around local minimizers, recently proposed in the context of Goldstein-subgradient-based algorithms and their behavior in practice. On generic examples, these algorithms are often observed to converge approximately linearly. We focus on objectives that are piecewise twice continuously differentiable. In that case, the new growth condition holds when the objective is, in addition, strongly convex. In the nonconvex case, we prove that quadratic growth in conjunction with a regularity property for the active gradients suffices. Computational experiments illustrate the importance of the assumptions.
