A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization

TL;DR

This paper introduces a Newton-CG based barrier-augmented Lagrangian method for efficiently finding approximate second-order stationary points in complex nonconvex conic optimization problems, with proven complexity bounds and superior numerical performance.

Contribution

It is the first to analyze the complexity of finding approximate SOSPs in general nonconvex conic optimization, providing new theoretical bounds and a novel algorithm.

Findings

Achieves $ ilde{O}( ext{epsilon}^{-11/2})$ inner iteration complexity.

Achieves $ ilde{O}( ext{epsilon}^{-11/2} imes ext{min}igrace n, ext{epsilon}^{-5/4}igrace)$ operation complexity.

Numerical results show the proposed method outperforms first-order methods.

Abstract

In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under some mild assumptions, we show that our method enjoys a total inner iteration complexity of $\widetilde{\cal O}(\epsilon^{-11/2})$ and an operation complexity of $\widetilde{\cal O}(\epsilon^{-11/2}\min\{n,\epsilon^{-5/4}\})$ for finding an $(\epsilon,\sqrt{\epsilon})$-SOSP of general nonconvex conic optimization with high probability. Moreover, under a constraint qualification, these complexity bounds are improved to $\widetilde{\cal O}(\epsilon^{-7/2})$ and $\widetilde{\cal O}(\epsilon^{-7/2}\min\{n,\epsilon^{-3/4}\})$, respectively. To the best of our knowledge, this is the first study on the complexity of finding an approximate SOSP of general nonconvex conic optimization. Preliminary numerical results are presented to demonstrate superiority of the proposed method over first-order methods in terms of solution quality.