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. SNAP is a second-order algorithm that requires O^e(max{1/ϵ²_G, 1/ϵ³_H}) iterations to compute an (ϵ_G, ϵ_H)-SOSP, where O^e hides the iteration complexity for eigenvalue-decomposition. Building on SNAP, we propose a first-order algorithm, named SNAP⁺, that requires O(1/ϵ^2.5) iterations to compute (ϵ, ^√ϵ)-SOSP. The per-iteration computational complexities of our algorithms are polynomial in the number of constraints and problem dimension. To the best of our knowledge, this is the first time that first-order algorithms with polynomial per-iteration complexity and global sublinear rate are designed to find SOSPs of the important class of non-convex problems with linear constraints (almost surely).