Exact Lipschitz Regularization of Convex Optimization Problems
针对一类凸复合最小化问题，提出一种无需非光滑目标函数Lipschitz连续假设的精确Lipschitz正则化框架，并利用该框架建立基于原始数据的函数值收敛速度。
Abstract We consider the class of convex composite minimization problems which consists of minimizing the sum of two nonsmooth extended valued convex functions, with one which is composed with a linear map. Convergence rate guarantees for first order methods on this class of problems often require the additional assumption of Lipschitz continuity of the nonsmooth objective function composed with the linear map. We introduce a theoretical framework where the restrictive Lipschitz continuity of this function is not required. Building on a novel dual representation of the so-called Pasch-Hausdorff envelope, we derive an exact Lipshitz regularization for this class of problems. We then show how the aforementioned result can be utilized in establishing function values-based rates of convergence in terms of the original data. Throughout, we provide examples and applications which illustrate the potential benefits of our approach.