Tomoki Ohsawa

Anthony Bloch, Leonardo Colombo, Rohit Gupta et al.

We investigate symmetry reduction of optimal control problems for left-invariant control systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuous-time formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler--Poincaré equations from a variational principle. By applying a Legendre transformation to it, we recover the Lie-Poisson equations obtained by A. D. Borum [Master's Thesis, University of Illinois at Urbana-Champaign, 2015] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie-Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle.

Molei Tao, Tomoki Ohsawa

The article considers smooth optimization of functions on Lie groups. By generalizing NAG variational principle in vector space (Wibisono et al., 2016) to Lie groups, continuous Lie-NAG dynamics which are guaranteed to converge to local optimum are obtained. They correspond to momentum versions of gradient flow on Lie groups. A particular case of $\mathsf{SO}(n)$ is then studied in details, with objective functions corresponding to leading Generalized EigenValue problems: the Lie-NAG dynamics are first made explicit in coordinates, and then discretized in structure preserving fashions, resulting in optimization algorithms with faithful energy behavior (due to conformal symplecticity) and exactly remaining on the Lie group. Stochastic gradient versions are also investigated. Numerical experiments on both synthetic data and practical problem (LDA for MNIST) demonstrate the effectiveness of the proposed methods as optimization algorithms ($not$ as a classification method).