Speaker: Researcher Xin Liu（Chinese Academy of Sciences）

Title: A Class of Smooth Exact Penalty Function Methods for Optimization Problems with Orthogonality Constraints

Time: Thurs Sept.18, 2019,PM:2:30-3:30

Location: Conference room, floor 6, Building No.3, Wushan Campus

Abstract：

     Updating the augmented Lagrangian multiplier by closed-form expression yields efficient first-order infeasible approach for optimization problems with orthogonality constraints. Hence, parallelization becomes tractable in solving this type of problems. Inspired by this closed-form updating scheme, we propose an exact penalty function model with compact convex constraints (PenC). We show its equivalence to optimization problems with orthogonality constraints under mild condition. Based on PenC, we first propose a first-order algorithm called PenCF and establish its global convergence and local linear convergence rate under some mild assumptions. If the computation and storage of Hessian is achievable, and we pursue high precision solution and fast local convergence rate, a second-order approach called PenCS is proposed under the same penalty function. To avoid expensive calculation or solving a hard subproblem in computing the Newton step, we propose a new strategy to do it approximately which leads to quadratic convergence theoretically. Moreover, the main iterations of both PenCF and PenCS are orthonormalization-free and hence parallelizable. Numerical experiments illustrate that PenCF is comparable with existing first-order methods including the existent infeasible approaches. Furthermore, PenCS shows its stability and high efficiency in obtain high precision solution in comparing with the existent second-order methods.