A restricted memory quasi-Newton bundle method for nonsmooth optimization on Riemannian manifolds

In this paper, a restricted memory quasi-Newton bundle method for minimizing a locally Lipschitz function over a Riemannian manifold is proposed, which extends the classical one in Euclidean spaces to the manifold setting. The curvature information of the objective function is approximated by applying the Riemannian version of the quasi-Newton updating formulas. The subgradient aggregation technique is used to avoid solving the time-consuming quadratic programming subproblem when calculating the candidate descent direction. Moreover, a new Riemannian line search procedure is proposed to generate the stepsizes, and the process is finitely terminated under a new version of the Riemannian semismooth assumption. Global convergence of the proposed method is established: if the serious iteration steps are finite, then the last serious iterate is stationary; otherwise, every accumulation point of the serious iteration sequence is stationary. Finally, some preliminary numerical results show that the proposed method is efficient

A simple version of bundle method with linear programming

Bundle methods have been well studied in nonsmooth optimization. In most of the bundle methods developed thus far (traditional bundle methods), at least one quadratic programming subproblem needs to be solved in each iteration. In this paper, a simple version of bundle method with linear programming is proposed. In each iteration, a cutting-plane model subject to a constraint constructed by an infinity norm is minimized. Without line search or trust region techniques, the convergence of the method can be shown. Additionally, the infinity norm in the constraint can be generalized to p-norm. Preliminary numerical experiments show the potential advantage of the proposed method for solving large scale problems2391412FAPESP – Fundação de Amparo à Pesquisa Do Estado De São Paulo2017/15936-2ARCFondation ARC pour la Recherche sur le CancerAustralian Research Council [DP12100567]; FAPESPFundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP) [2017/15936-2