A Newton-CG based barrier method for finding a second-order stationary point of nonconvex conic optimization with complexity guarantees

TL;DR

This paper introduces a Newton-CG based barrier method for nonconvex conic optimization that efficiently finds approximate second-order stationary points with proven iteration and operation complexity guarantees.

Contribution

It develops a novel barrier method with complexity guarantees for finding second-order stationary points in nonconvex conic optimization, matching the best known iteration bounds.

Findings

Achieves ${ m O}( ext{epsilon}^{-3/2})$ iteration complexity.

Establishes operation complexity involving Cholesky factorizations and fundamental operations.

Matches the best known complexity bounds for second-order methods in unconstrained nonconvex optimization.

Abstract

In this paper we consider finding an approximate second-order stationary point (SOSP) of nonconvex conic optimization that minimizes a twice differentiable function over the intersection of an affine subspace and a convex cone. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier method for finding an $(\epsilon,\sqrt{\epsilon})$-SOSP of this problem. Our method is not only implementable, but also achieves an iteration complexity of ${\cal O}(\epsilon^{-3/2})$, which matches the best known iteration complexity of second-order methods for finding an $(\epsilon,\sqrt{\epsilon})$-SOSP of unconstrained nonconvex optimization. The operation complexity, consisting of ${\cal O}(\epsilon^{-3/2})$ Cholesky factorizations and $\widetilde{\cal O}(\epsilon^{-3/2}\min\{n,\epsilon^{-1/4}\})$ other fundamental operations, is also established for our method.