Session: Feasible and infeasible methods for optimization on manifolds I
Chair: Bin Gao
Cluster: Optimization on Manifolds
Talk 1: A double tracking method for optimization with decentralized generalized orthogonality constraints
Speaker: Xin Liu
Abstract: We consider the decentralized optimization problems with generalized orthogonality constraints, where both the objective function and the constraint exhibit a distributed structure. Such optimization problems, albeit ubiquitous in practical applications, remain unsolvable by existing algorithms in the presence of distributed constraints. To address this issue, we convert the original problem into an unconstrained penalty model by resorting to the recently proposed constraint-dissolving operator. However, this transformation compromises the essential property of separability in the resulting penalty function, rendering it impossible to employ existing algorithms to solve. We overcome this difficulty by introducing a novel algorithm that tracks the gradient of the objective function and the Jacobian of the constraint mapping simultaneously. The global convergence guarantee is rigorously established with an iteration complexity. To substantiate the effectiveness and efficiency of our proposed algorithm, we present numerical results on both synthetic and real-world datasets.
Talk 2: Constrained Saddle Dynamics for Index-1 Saddle Point Search on Riemannian Manifolds
Speaker: Yukuan Hu
Abstract: Index-1 saddle point search on Riemannian manifolds is a fundamental task in various applications. However, most existing works concentrate on unconstrained settings, with only limited efforts given to the special cases where the manifolds are induced by global defining functions. In this talk, we introduce a constrained saddle dynamics applicable to general Riemannian submanifolds embedded in Euclidean spaces, where the position and direction variables are simultaneously evolved on the tangent bundle. In particular, the direction dynamics leverages the second fundamental form to ensure feasibility. The linear stability of the dynamics at index-1 saddle points is established. We further discretize the proposed dynamics using tools from Riemannian geometry. The local convergence properties of the resulting iterative methods are analyzed. Finally, the numerical experiments on electronic excited states calculations demonstrate the effectiveness of the proposed methods.
Talk 3: Efficient optimization with orthogonality constraints via random submanifold approach
Speaker: Andi Han
Abstract: Optimization problems with orthogonality constraints are commonly addressed using Riemannian optimization, which leverages the geometric structure of the constraint set as a Riemannian manifold. This method involves computing a search direction in the tangent space and updating via a retraction. However, the computational cost of the retraction increases with problem size. To improve scalability, we propose a method that restricts updates to random submanifolds, reducing per-iteration complexity. We introduce two submanifold selection strategies and analyze the convergence for nonconvex functions, including those satisfying the Riemannian Polyak–Łojasiewicz condition, as well as for stochastic optimization problems. The approach generalizes to quotient manifolds derived from the orthogonal manifold.
