Worst case complexity of direct search under convexity

• 成果类型: Article
• 署名作者: Dodangeh, M.; Vicente, L. N.
• 署名单位: Universidade de Coimbra; Universidade de Coimbra
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0847-0
• 发表日期: 2016
• 页码: 307-332
• 关键词: Optimization; descent

摘要

In this paper we prove that the broad class of direct-search methods of directional type, based on imposing sufficient decrease to accept new iterates, exhibits the same worst case complexity bound and global rate of the gradient method for the unconstrained minimization of a convex and smooth function. More precisely, it will be shown that the number of iterations needed to reduce the norm of the gradient of the objective function below a certain threshold is at most proportional to the inverse of the threshold. It will be also shown that the absolute error in the function values decay at a sublinear rate proportional to the inverse of the iteration counter. In addition, we prove that the sequence of absolute errors of function values and iterates converges r-linearly in the strongly convex case.