Hessian barrier algorithms for linearly constrained optimization problems

TL;DR

This paper introduces the Hessian barrier algorithm (HBA), an interior-point method for linearly constrained optimization that converges to critical points and has a quantifiable convergence rate, validated through numerical experiments.

Contribution

The paper presents HBA, a novel interior-point method combining Hessian Riemannian gradient flows with backtracking, unifying several existing methods and providing convergence guarantees.

Findings

Converges to critical points under non-degeneracy.

Global convergence to minima in convex cases.

Convergence rate of O(1/k^ρ) for quadratic programs.

Abstract

In this paper, we propose an interior-point method for linearly constrained optimization problems (possibly nonconvex). The method - which we call the Hessian barrier algorithm (HBA) - combines a forward Euler discretization of Hessian Riemannian gradient flows with an Armijo backtracking step-size policy. In this way, HBA can be seen as an alternative to mirror descent (MD), and contains as special cases the affine scaling algorithm, regularized Newton processes, and several other iterative solution methods. Our main result is that, modulo a non-degeneracy condition, the algorithm converges to the problem's set of critical points; hence, in the convex case, the algorithm converges globally to the problem's minimum set. In the case of linearly constrained quadratic programs (not necessarily convex), we also show that the method's convergence rate is $\mathcal{O}(1/k^\rho)$ for some $\rho\in(0,1]$ that depends only on the choice of kernel function (i.e., not on the problem's primitives). These theoretical results are validated by numerical experiments in standard non-convex test functions and large-scale traffic assignment problems.