A Sequential Cubic Programming Method with Second-Order Complexity Guarantees for Equality Constrained Optimization

TL;DR

This paper introduces a new sequential cubic programming method for equality constrained optimization that guarantees second-order convergence and provides the best known worst-case complexity bounds.

Contribution

It develops a novel algorithm with second-order complexity guarantees and demonstrates its global convergence and local quadratic convergence properties.

Findings

First method with worst-case complexity guarantees for this problem class.

Achieves complexity bounds of O(εg^{-3/2}), O(εH^{-3}), and O(εc^{-1}).

Ensures global convergence to second-order stationary points.

Abstract

We develop a new method for equality constrained optimization problems based on a sequential cubic programming framework. Each iteration utilizes a step decomposition based on the Jacobian of the constraints into a normal and a tangential component, the latter of which is found by solving a subproblem involving cubic regularization. The method incorporates second-order correction steps as necessary to ensure global convergence to second-order stationary points as well as local quadratic convergence. In addition, we show that the algorithm is the first to obtain worst case complexity guarantees on the order of $\mathcal{O}(\epsilon_g^{-3/2})$ for the gradient of the Lagrangian, $\mathcal{O}(\epsilon_H^{-3})$ in terms of second-order stationarity, and $\mathcal{O}(\epsilon_c^{-1})$ in terms of the constraint violation. These are the best known complexity guarantees of any method for this class of problems.