Global convergence of the gradient method for functions definable in o-minimal structures

• 成果类型: Article
• 署名作者: Josz, Cedric
• 署名单位: Columbia University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01937-5
• 发表日期: 2023
• 页码: 355-383
• 关键词: lojasiewicz inequalities; 1st-order methods; descent methods; real field; minimization; expansions; continuity; nonconvex; CURVES; sets

摘要

We consider the gradient method with variable step size for minimizing functions that are definable in o-minimal structures on the real field and differentiable with locally Lipschitz gradients. We prove that global convergence holds if continuous gradient trajectories are bounded, with the minimum gradient norm vanishing at the rate o(1/k) if the step sizes are greater than a positive constant. If additionally the gradient is continuously differentiable, all saddle points are strict, and the step sizes are constant, then convergence to a local minimum holds almost surely over any bounded set of initial points.