Generalized Self-concordant Hessian-barrier algorithms

TL;DR

This paper introduces a new interior-point algorithm based on generalized self-concordant functions for optimizing challenging non-convex problems with boundary singularities, demonstrating global convergence and optimal complexity.

Contribution

It extends Hessian-barrier methods to a broader class of functions, providing theoretical guarantees and practical applications in statistical estimation and $L^{p}$-minimization.

Findings

Proves global convergence to approximate stationary points.

Establishes worst-case optimal iteration complexity.

Demonstrates efficiency in applications like statistical estimation.

Abstract

Many problems in statistical learning, imaging, and computer vision involve the optimization of a non-convex objective function with singularities at the boundary of the feasible set. For such challenging instances, we develop a new interior-point technique building on the Hessian-barrier algorithm recently introduced in Bomze, Mertikopoulos, Schachinger and Staudigl, [SIAM J. Opt. 2019 29(3), pp. 2100-2127], where the Riemannian metric is induced by a generalized self-concordant function. This class of functions is sufficiently general to include most of the commonly used barrier functions in the literature of interior point methods. We prove global convergence to an approximate stationary point of the method, and in cases where the feasible set admits an easily computable self-concordant barrier, we verify worst-case optimal iteration complexity of the method. Applications in non-convex statistical estimation and $L^{p}$-minimization are discussed to given the efficiency of the method.