Constrained Optimization on Matrix Lie Groups via Interior-Point Method

TL;DR

This paper introduces MLG-IPM, an interior-point method for optimization on matrix Lie groups that operates directly on the group structure, ensuring intrinsic feasibility and improved performance over existing Riemannian methods.

Contribution

The paper develops a novel interior-point framework for matrix Lie group optimization that avoids redundant representations and Riemannian metrics, with proven quadratic convergence and superior empirical results.

Findings

Achieves higher success rates than Riemannian methods.

Requires fewer iterations for convergence.

Demonstrates superior numerical accuracy and robustness.

Abstract

This paper proposes an interior-point framework for constrained optimization problems whose decision variables evolve on matrix Lie groups. The proposed method, termed the Matrix Lie Group Interior-Point Method (MLG-IPM), operates directly on the group structure using a minimal Lie algebra parametrization, avoiding redundant matrix representations and eliminating explicit dependence on Riemannian metrics. A primal-dual formulation is developed in which the Newton system is constructed through sensitivity and curvature matrices. Also, multiplicative updates are performed via the exponential map, ensuring intrinsic feasibility with respect to the group structure while maintaining strict positivity of slack and dual variables through a barrier strategy. A local analysis establishes quadratic convergence under standard regularity assumptions and characterizes the behavior under inexact Newton steps. Statistical comparisons against Riemannian Interior-Point Methods, specifically for optimization problems defined over the Special Orthogonal Group SO(n) and Special Linear Group SL(n), demonstrate that the proposed approach achieves higher success rates, fewer iterations, and superior numerical accuracy. Furthermore, its robustness under perturbations suggests that this method serves as a consistent and reliable alternative for structured manifold optimization.