Sequential Quadratic Optimization for Nonlinear Optimization Problems on Riemannian Manifolds

TL;DR

This paper introduces RSQO, a novel Riemannian sequential quadratic optimization algorithm with proven global and local convergence, effectively solving constrained nonlinear problems on manifolds with superior stability and accuracy.

Contribution

The paper develops the first algorithm with both global and local convergence guarantees for Riemannian constrained optimization problems.

Findings

RSQO achieves higher accuracy than existing methods.

RSQO demonstrates stable convergence in empirical tests.

Theoretical proofs establish global and local convergence properties.

Abstract

We consider optimization problems on Riemannian manifolds with equality and inequality constraints, which we call Riemannian nonlinear optimization (RNLO) problems. Although they have numerous applications, the existing studies on them are limited especially in terms of algorithms. In this paper, we propose Riemannian sequential quadratic optimization (RSQO) that uses a line-search technique with an ell_1 penalty function as an extension of the standard SQO algorithm for constrained nonlinear optimization problems in Euclidean spaces to Riemannian manifolds. We prove its global convergence to a Karush-Kuhn-Tucker point of the RNLO problem by means of parallel transport and the exponential mapping. Furthermore, we establish its local quadratic convergence by analyzing the relationship between sequences generated by RSQO and the Riemannian Newton method. Ours is the first algorithm that has both global and local convergence properties for constrained nonlinear optimization on Riemannian manifolds. Empirical results show that RSQO finds solutions more stably and with higher accuracy compared with the existing Riemannian penalty and augmented Lagrangian methods.