AI 中文总结
本文提出一种用于希尔伯特空间非凸可微函数极小化的类干摩擦非单调线搜索快速梯度算法，可实现不同收敛性，能退化现有算法，适用于弱凸非光滑优化，仿真验证其有效性。
AI 中文摘要
本文针对希尔伯特空间中可微（可能非凸）函数的极小化问题，提出一种快速梯度算法。首先将凸函数的干摩擦性质推广到非凸情形下的类干摩擦性质，随后采用线搜索技术在每次迭代中自适应更新参数。根据参数选择的不同，所提算法可实现子序列收敛到临界点，或完全序列收敛到目标函数的“近似”临界点；在 merit 函数的 Kurdyka–Lojasiewicz（KL）性质下，还可建立完全序列收敛到临界点的结论。得益于参数的灵活性，该算法可退化为多种现有的带 Hessian 阻尼和干摩擦的惯性梯度算法。通过利用 Moreau 包络的变分性质，所提算法可用于求解弱凸非光滑优化问题，尤其将凸 KL 函数的 Moreau 包络的 KL 指数结果推广到一大类非凸且未必连续的 KL 函数。仿真结果验证了该算法的有效性，并展现了类干摩擦与外推、线搜索技术结合的潜在优势。
英文摘要
In this paper, we propose a fast gradient algorithm for the problem of minimizing a differentiable (possibly nonconvex) function in Hilbert spaces. We first extend the dry friction property for convex functions to what we call the dry-like friction property in a nonconvex setting, and then employ a line search technique to adaptively update parameters at each iteration. Depending on the choice of parameters, the proposed algorithm exhibits subsequential convergence to a critical point or full sequential convergence to an ``approximate'' critical point of the objective function. We also establish the full sequential convergence to a critical point under the Kurdyka--Lojasiewicz (KL) property of a merit function. Thanks to the parameters' flexibility, our algorithm can reduce to a number of existing inertial gradient algorithms with Hessian damping and dry friction. By exploiting variational properties of the Moreau envelope, the proposed algorithm is adapted to address weakly convex nonsmooth optimization problems. In particular, we extend the result on KL exponent for the Moreau envelope of a convex KL function to a broad class of KL functions that are not necessarily convex nor continuous. Simulation results illustrate the efficiency of our algorithm and demonstrate the potential advantages of combining dry-like friction with extrapolation and line search techniques.
Journal refSIAM Journal on Optimization, 34(3) 2557-2587, 2024
