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: A projected gradient descent algorithm for ab initio fixed-volume crystal structure relaxation
Speaker: Yukuan Hu
Abstract: This paper is concerned with ab initio crystal structure relaxation under a fixed unit cell volume, which is a step in calculating the static equations of state and forms the basis of thermodynamic property calculations for materials. The task can be formulated as an energy minimization with a determinant constraint. Widely used line minimization-based methods (e.g., conjugate gradient method) lack both efficiency and convergence guarantees due to the nonconvex nature of the determinant constraint as well as the significant differences in the curvatures of the potential energy surface with respect to atomic and lattice components. To this end, we propose a projected gradient descent algorithm named PANBB. It is equipped with (i) search direction projections for lattice vectors, (ii) distinct curvature-aware initial trial step sizes for atomic and lattice updates, and (iii) a nonrestrictive line minimization criterion as the stopping rule for the inner loop. It can be proved that PANBB favors theoretical convergence to equilibrium states. Across a benchmark set containing 223 structures from various categories, PANBB achieves average speedup factors of approximately 1.41 and 1.45 over the conjugate gradient method and direct inversion in the iterative subspace implemented in off-the-shelf simulation software, respectively. Moreover, it normally converges on all the systems, manifesting its robustness. As an application, we calculate the static equations of state for the high-entropy alloy AlCoCrFeNi, which remains elusive owing to 160 atoms representing both chemical and magnetic disorder and the strong local lattice distortion. The results are consistent with the previous calculations and are further validated by experimental thermodynamic data.
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.
