The Computational Complexity of Avoiding Strict Saddle Points in Constrained Optimization

2026-04-02

AI Summary

This study resolves a long-standing open problem regarding the computational complexity of finding approximate second-order stationary points (SOSPs) in constrained non-convex optimization. By integrating the Karush–Kuhn–Tucker (KKT) conditions with second-order optimality criteria and leveraging complexity classes such as PLS, PPAD, and CLS, the authors establish for the first time that the constrained SOSP problem remains PLS-complete—even when restricted to the two-dimensional unit square and when stable points lie strictly in the interior. This result implies that, unless PLS ⊆ PPAD, no efficient deterministic continuous iterative algorithm can exist for this task. The work not only delineates the inherent theoretical hardness of constrained SOSP computation but also presents the first PLS-complete problem instance within a compact domain that strictly generalizes Real-LocalOpt.

Abstract

While first-order stationary points (FOSPs) are the traditional targets of non-convex optimization, they often correspond to undesirable strict saddle points. To circumvent this, attention has shifted towards second-order stationary points (SOSPs). In unconstrained settings, finding approximate SOSPs is PLS-complete (Kontogiannis et al.), matching the complexity of finding unconstrained FOSPs (Hollender and Zampetakis). However, the complexity of finding SOSPs in constrained settings remained notoriously unclear and was highlighted as an important open question by both aforementioned works. Under one strict definition, even verifying whether a point is an approximate SOSP is NP-hard (Murty and Kabadi). Under another widely adopted, relaxed definition where non-negative curvature is required only along the null space of the active constraints, the problem lies in TFNP, and algorithms with O(poly(1/epsilon)) running times have been proposed (Lu et al.). In this work, we settle the complexity of constrained SOSP by proving that computing an epsilon-approximate SOSP under the tractable definition is PLS-complete. We demonstrate that our result holds even in the 2D unit square [0,1]^2, and remarkably, even when stationary points are isolated at a distance of Omega(1) from the domain's boundary. Our result establishes a fundamental barrier: unless PLS is a subset of PPAD (implying PLS = CLS), no deterministic, iterative algorithm with an efficient, continuous update rule can exist for finding approximate SOSPs. This contrasts with the constrained first-order counterpart, for which Fearnley et al. showed that finding an approximate KKT point is CLS-complete. Finally, our result yields the first problem defined in a compact domain to be shown PLS-complete beyond the canonical Real-LocalOpt (Daskalakis and Papadimitriou)."

Problem

• constrained optimization
• second-order stationary points
• computational complexity
• PLS-completeness
• approximate SOSP

Innovation

• PLS-completeness
• constrained optimization
• second-order stationary points
• computational complexity
• TFNP