Convergence of augmented Lagrangian methods in extensions beyond nonlinear programming

• 成果类型: Article
• 署名作者: Rockafellar, R. Tyrrell
• 署名单位: University of Washington; University of Washington Seattle
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01832-5
• 发表日期: 2023
• 页码: 375-420
• 关键词: proximal point algorithm; local convergence; optimization

摘要

The augmented Lagrangian method (ALM) is extended to a broader-than-ever setting of generalized nonlinear programming in convex and nonconvex optimization that is capable of handling many common manifestations of nonsmoothness. With the help of a recently developed sufficient condition for local optimality, it is shown to be derivable from the proximal point algorithm through a kind of local duality corresponding to an optimal solution and accompanying multiplier vector that furnish a local saddle point of the augmented Lagrangian. This approach leads to surprising insights into stepsize choices and new results on linear convergence that draw on recent advances in convergence properties of the proximal point algorithm. Local linear convergence is shown to be assured for a class of model functions that covers more territory than before.