Hessian barrier algorithms for non-convex conic optimization

TL;DR

This paper introduces a new class of adaptive Hessian barrier algorithms for non-convex conic optimization, achieving optimal worst-case iteration complexity for first- and second-order methods under weak conditions.

Contribution

It develops a novel potential-reduction based approach using self-concordant barriers, attaining optimal complexity bounds for general conic constrained problems.

Findings

Achieves $O( ext{epsilon}^{-2})$ iteration complexity for first-order methods.

Achieves $O( ext{epsilon}^{-3/2})$ iteration complexity for second-order methods.

First method to attain these bounds under weak conditions for non-convex conic problems.

Abstract

We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first- and second-order optimization methods, building on the Hessian-barrier techniques introduced in [Bomze, Mertikopoulos, Schachinger, and Staudigl, Hessian barrier algorithms for linearly constrained optimization problems, SIAM Journal on Optimization, 2019]. Our approach is based on a potential-reduction mechanism and attains a suitably defined class of approximate first- or second-order KKT points with the optimal worst-case iteration complexity $O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively. A key feature of our methodology is the use of self-concordant barrier functions to construct strictly feasible iterates via a disciplined decomposition approach and without sacrificing on the iteration complexity of the method. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained optimization problems.