A Dynamic Smoothing Technique for a Class of Nonsmooth Optimization Problems on Manifolds
提出一种动态平滑梯度下降算法(DSGM)，用于求解流形上光滑非凸与非光滑凸函数之和的最小化问题，证明了所有聚点均为稳定点，并给出了O(1/k^{1/3})的收敛速率。
.We consider the problem of minimizing the sum of a smooth nonconvex function and a nonsmooth convex function over a compact embedded submanifold. We describe an algorithm, which we refer to as "dynamic smoothing gradient descent on manifolds" (DSGM), that is based on applying Riemmanian gradient steps on a series of smooth approximations of the objective function that are determined by a diminishing sequence of smoothing parameters. The DSGM algorithm is simple and can be easily employed for a broad class of problems without any complex adjustments. We show that all accumulation points of the sequence generated by the method are stationary. We devise a convergence rate of \(O(\frac{1}{k^{1/3}})\) in terms of an optimality measure that can be easily computed. Numerical experiments illustrate the potential of the DSGM method.Keywordsmanifold optimizationdynamic smoothingrate of convergencefirst-order methodsMSC codes90C26