An Invariant Set Approach for Optimization on Integrable Manifolds

TL;DR

This paper introduces an invariant set approach that transforms constrained optimization problems on nonlinear manifolds into unconstrained problems by leveraging invariant set theory, facilitating easier solutions.

Contribution

It proposes a novel algorithm that applies invariant set theory to optimize on integrable manifolds, bridging control theory and numerical optimization.

Findings

Successfully lifts manifold problems to vector spaces

Transforms constrained problems into unconstrained ones

Provides a new framework for manifold optimization

Abstract

Recent results in control systems and numerical integration literature utilize invariant set theory to lift dynamical systems evolving on nonlinear manifolds to those evolving on vector spaces. We leverage this technique to propose an algorithm to solve a class of constrained optimization problems as unconstrained problems.