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Riemannian Interior Point Methods for Constrained Optimization on Manifolds

Zhijian Lai, Akiko Yoshise

TL;DR

This work extends the classical primal-dual interior point method from the Euclidean setting to the Riemannian one, and establishes its local superlinear and quadratic convergence under the standard assumptions.

Abstract

We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method, is for solving Riemannian constrained optimization problems. We establish its local superlinear and quadratic convergence under the standard assumptions. Moreover, we show its global convergence when it is combined with a classical line search. Our method is a generalization of the classical framework of primal-dual interior point methods for nonlinear nonconvex programming. Numerical experiments show the stability and efficiency of our method.

Riemannian Interior Point Methods for Constrained Optimization on Manifolds

TL;DR

This work extends the classical primal-dual interior point method from the Euclidean setting to the Riemannian one, and establishes its local superlinear and quadratic convergence under the standard assumptions.

Abstract

We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method, is for solving Riemannian constrained optimization problems. We establish its local superlinear and quadratic convergence under the standard assumptions. Moreover, we show its global convergence when it is combined with a classical line search. Our method is a generalization of the classical framework of primal-dual interior point methods for nonlinear nonconvex programming. Numerical experiments show the stability and efficiency of our method.
Paper Structure (29 sections, 21 theorems, 73 equations, 3 tables, 5 algorithms)

This paper contains 29 sections, 21 theorems, 73 equations, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Riemannian Geometry
  4. Riemannian Newton Method
  5. Description of Riemannian Interior Point Methods
  6. KKT Vector Field and Its Covariant Derivative
  7. Implication of Standard Assumptions
  8. Prototype Algorithm of RIPM
  9. Solving Perturbed Newton Equation Efficiently
  10. Condensed Form of Perturbed Newton Equation
  11. Krylov Subspace Methods on Tangent Space
  12. Preliminaries and Auxiliary Results
  13. Vector Transport and Parallel Transport
  14. Lipschitz Continuity with respect to Vector Transports
  15. Auxiliary Lemmas
  16. ...and 14 more sections

Key Result

Theorem 3.1

If the standard Riemannian assumptions B1-B5 hold at some point $w^{*}$, then the operator $\nabla F(w^{*})$ in (eq:CovariantDerivativeKKTVectorField) is nonsingular.

Theorems & Definitions (38)

  • Theorem 3.1
  • proof
  • Remark 4.1
  • Definition 4.2: huang2013optimization
  • Definition 4.3: huang2015riemannian
  • Lemma 4.4: huang2015riemannian
  • Definition 4.5
  • Lemma 4.6
  • Lemma 4.7: huang2015riemannian
  • Lemma 4.8: huang2015riemannian
  • ...and 28 more