In this paper we develop accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient (LLCG), which is beyond the well-studied class of convex optimization with Lipschitz continuous gradient. In particular, we first consider unconstrained convex optimization with LLCG and propose accelerated proximal gradient (APG) methods for solving it. The proposed APG methods are equipped with a verifiable termination criterion and enjoy an operation complexity of ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ and ${\cal O}(\log \varepsilon^{-1})$ for finding an $\varepsilon$-residual solution of an unconstrained convex and strongly convex optimization problem, respectively. We then consider constrained convex optimization with LLCG and propose an first-order proximal augmented Lagrangian method for solving it by applying one of our proposed APG methods to approximately solve a sequence of proximal augmented Lagrangian subproblems. The resulting method is equipped with a verifiable termination criterion and enjoys an operation complexity of ${\cal O}(\varepsilon^{-1}\log \varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ for finding an $\varepsilon$-KKT solution of a constrained convex and strongly convex optimization problem, respectively. All the proposed methods in this paper are parameter-free or almost parameter-free except that the knowledge on convexity parameter is required. In addition, preliminary numerical results are presented to demonstrate the performance of our proposed methods. To the best of our knowledge, no prior studies were conducted to investigate accelerated first-order methods with complexity guarantees for convex optimization with LLCG. All the complexity results obtained in this paper are new.

翻译：本文针对梯度局部Lipschitz连续（LLCG）的凸优化问题，提出了加速一阶方法，突破了传统梯度全局Lipschitz连续凸优化的研究范畴。首先，考虑无约束LLCG凸优化问题，提出加速近端梯度（APG）方法进行求解。所提APG方法配备可验证的终止准则，在寻找无约束凸优化问题的$\varepsilon$-残差解时具有${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$的操作复杂度，而对于强凸优化问题则达到${\cal O}(\log \varepsilon^{-1})$复杂度。继而，针对约束LLCG凸优化问题，提出一阶近端增广拉格朗日方法，通过将所提APG方法之一应用于近似求解一系列近端增广拉格朗日子问题来实现求解。该方法同样配备可验证终止准则，在寻找约束凸优化问题的$\varepsilon$-KKT解时具有${\cal O}(\varepsilon^{-1}\log \varepsilon^{-1})$复杂度，而对于约束强凸优化问题则达到${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$复杂度。本文所有方法均为无参数或近乎无参数（除需已知凸性参数外）。此外，给出初步数值实验结果以验证所提方法的性能。据我们所知，此前尚无研究探讨具有复杂度保证的LLCG凸优化加速一阶方法。本文所有复杂度结果均为新成果。