Barrier Algorithms for Constrained Non-Convex Optimization

TL;DR

This paper extends interior-point methods with barrier functions to non-convex constrained optimization, achieving favorable global complexity bounds similar to unconstrained problems.

Contribution

It introduces first- and second-order algorithms for non-convex problems with convex constraints, with iteration complexity comparable to unconstrained optimization.

Findings

Methods attain approximate KKT points with $O( ext{epsilon}^{-2})$ and $O( ext{epsilon}^{-3/2})$ complexity.

Theoretical analysis shows favorable global complexity beyond convex optimization.

Proposes algorithms applicable to general convex set and linear constraints.

Abstract

In this paper we theoretically show that interior-point methods based on self-concordant barriers possess favorable global complexity beyond their standard application area of convex optimization. To do that we propose first- and second-order methods for non-convex optimization problems with general convex set constraints and linear constraints. Our methods attain a suitably defined class of approximate first- or second-order KKT points with the worst-case iteration complexity similar to unconstrained problems, namely $O(\varepsilon^{-2})$ (first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively.