In this talk, we consider a class of orthogonal constrained optimization problems, the feasible region of which is called the Stiefel manifold. Our new proposed framework combines a function value reduction step with a multiplier symmetrization step. Different with the existing approaches, the function value reduction is conducted in the Euclidean space instead of the Stiefel manifold or its tangent space. We construct two types of algorithms based on this new framework. The first type is gradient reduction based algorithms which consists of gradient reflection (GR) and gradient projection (GP) two implementations. The other one adopts a column-wise block coordinate descent (CBCD) scheme with a novel idea for solving the corresponding CBCD subproblem inexactly. Theoretically, we can prove that both GR/GP with a fixed stepsize and CBCD belong to our framework, and any clustering point of the iterates generated by the proposed framework is a first-order stationary point. Preliminary experiments illustrate that our new framework is of great potential.

刘歆副研究员2004年在北京大学获得本科学位, 2009年在中国科学院计算数学与科学工程计算研究所获得博士学位, 并留所工作至今。他主要从事最优化方法相关理论及其应用的研究, 包括特征值问题、矩阵分解、大数据、电子结构计算等。基于他的出色工作, 刘歆副研究员获得了诸多奖励, 包括基金委优秀青年基金、中国科学院数学与系统科学研究院陈景润未来之星等。