Finding Second-Order Stationary Points Efficiently in Smooth Nonconvex Linearly Constrained Optimization Problems

This paper proposes two efficient algorithms for computing approximate second-order stationary points (SOSPs) of problems with generic smooth non-convex objective functions and generic linear constraints. While finding (approximate) SOSPs for the class of smooth non-convex linearly constrained problems is computationally intractable, we show that generic problem instances in this class can be solved efficiently. Specifically, for a generic problem instance, we show that certain strict complementarity (SC) condition holds for all Karush-Kuhn-Tucker (KKT) solutions. Based on this condition, we design an algorithm named Successive Negative-curvature grAdient Projection (SNAP), which performs either conventional gradient projection or some negative curvature-based projection steps to find SOSPs. Building on SNAP, we propose a first-order algorithm, named SNAP, that requires \mathcal{O}(1/\epsilon {2.5}) iterations to compute (\epsilon, \sqrt{\epsilon}) -SOSP. The per-iteration computational complexities of our algorithms are polynomial in the number of constraints and problem dimension.