Nonsmooth nonconvex optimization on Riemannian manifolds via bundle trust region algorithm

• Published in: Computational optimization and applications 2024-07, Vol.88 (3), p.871-902
• Main Authors: Hoseini Monjezi, N., Nobakhtian, S., Pouryayevali, M. R.
• Format: Article
• Language: English
• Subjects: Applied mathematics; Convex and Discrete Geometry; Iterative algorithms; Management Science; Mathematics; Mathematics and Statistics; Methods; Operations Research; Operations Research/Decision Theory; Optimization; Optimization algorithms; Optimization techniques; Riemann manifold; Statistics; Topological manifolds
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-024-00569-5

This paper develops an iterative algorithm to solve nonsmooth nonconvex optimization problems on complete Riemannian manifolds. The algorithm is based on the combination of the well known trust region and bundle methods. According to the process of the most bundle methods, the objective function is approximated by a piecewise linear working model which is updated by adding cutting planes at unsuccessful trial steps. Then at each iteration, by solving a subproblem that employs the working model in the objective function subject to the trust region, a candidate descent direction is obtained. We study the algorithm from both theoretical and practical points of view and its global convergence is verified to stationary points for locally Lipschitz functions. Moreover, in order to demonstrate the reliability and efficiency, a MATLAB implementation of the proposed algorithm is prepared and results of numerical experiments are reported.