Dissolving Constraints for Riemannian Optimization

TL;DR

This paper introduces a novel constraint dissolving function for Riemannian optimization, enabling unconstrained minimization with preserved optimality properties and leveraging existing unconstrained optimization results.

Contribution

It proposes a new class of constraint dissolving functions with easy-to-compute gradients and Hessians, bridging unconstrained and Riemannian optimization.

Findings

CDF shares stationary points with the original problem

Convergence properties are inherited from unconstrained optimization

Illustrative examples demonstrate effectiveness

Abstract

In this paper, we consider optimization problems over closed embedded submanifolds of $\mathbb{R}^n$, which are defined by the constraints $c(x) = 0$. We propose a class of constraint dissolving approaches for these Riemannian optimization problems. In these proposed approaches, solving a Riemannian optimization problem is transferred into the unconstrained minimization of a constraint dissolving function named CDF. Different from existing exact penalty functions, the exact gradient and Hessian of CDF are easy to compute. We study the theoretical properties of CDF and prove that the original problem and CDF have the same first-order and second-order stationary points, local minimizers, and {\L}ojasiewicz exponents in a neighborhood of the feasible region. Remarkably, the convergence properties of our proposed constraint dissolving approaches can be directly inherited from the existing rich results in unconstrained optimization. Therefore, the proposed constraint dissolving approaches build up short cuts from unconstrained optimization to Riemannian optimization. Several illustrative examples further demonstrate the potential of our proposed constraint dissolving approaches.